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ANALYTIC   GEOMETRY 


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THE  MACMILLAN  COMPANY 

NEW  YORK  •    BOSTON   •   CHICAGO 
SAN   FRANCISCO 

MACMILLAN  &  CO.,  Limited 

LONDON   •    BOMBAY   •    CALCUTTA 
MELBOURNE 

THE  MACMILLAN  CO.  OF  CANADA,  Ltd. 

TORONTO 


ANALYTIC    GEOMETRY 


BY 


N.    C.    RIGGS,   M.S. 

ASSOCIATE   PROFESSOR  OP  APPLIED  MECHANICS 

FORMERLY   ASSISTANT   PROFESSOR   OF   MATHEMATICS 

CARNEGIE   TECHNICAL   SCHOOLS,   PITTSBURGH,   PA. 


Neto  gork 

THE   MACMILLAN  COMPANY 

1916 

All  rights  reserved 


COPTBIGHT,    1910, 

By  the   MACMILLAN  COMPANY. 


Set  up  and  electrotyped.     Published  September,  1910.     Reprinted 
January,  March,  September,  1911  ;   February,  July,  1912  ;  September, 
1913  ;  February,  July,  1914;  August,  1915 ;  August,  1916. 


NorixiooU  l$xn% 

J.  S.  Gushing  Co.  —  Berwick  &  Smith  Co. 

Norwood,  Mass.,  U.S.A. 


:^ 


^  PREFACE 

In  the  preparation  of  this  book  the  author  has  tried  to  keep 
in  mind  the  twofold  requirement  of  a  text-book  on  Analytic 
Geometry :  to  bring  out  clearly  the  fundamental  principles  and 
methods  of  the  subject,  and  to  make  it  a  natural  introduction 
to  more  advanced  work.  Since  for  most  students  of  Analytic 
Geometry  the  subject  is  quite  as  essential  as  a  preparation  for 
the  study  of  Calculus  as  it  is  valuable  for  its  own  methods  and 
body  of  facts,  the  method  and  notation  of  the  Calculus  have 
been  used  in  their  application  to  tangents,  normals,  and  maxima 
and  minima  in  the  plane,  and  to  tangent  planes  and  lines  in 
space. 

The  conic  sections  have  not  been  accorded  as  much  space 
relatively  as  in  most  text-books  on  the  subject,  but  it  is 
believed  that  the  student's  time  in  the  usual  brief  course  can 
be  spent  to  greater  profit  in  the  study  of  such  chapters  as 
those  on  Trigonometric  and  Exponential  Functions,  Parametric 
Equations,  Empirical  Equations,  Maxima  and  Minima,  and 
Graphical  Solution  of  Equations,  than  upon  a  prolonged  course 
on  the  conies.     Especially  is  this  true  for  engineering  students. 

The  answers  to  many  of  the  problems  have  not  been  given. 
Where  the  student  can  check  the  answer  by  graphical  means, 
it  is  best  that  he  should  thus  test  the  correctness  of  his  work, 
and  a  complete  list  of  answers  tends  to  take  away  his  incen- 
tive for  doing  this. 

The  author  is  under  many  obligations  to  Professors  D.  F. 
Campbell,  Alexander  Pell,  C.  W.  Leigh,  and  C.  I.  Palmer,  of 
the  Armour  Institute  of  Technology,  and  to  Mr.  Paul  Dorweiler 


vi  PREFACE 

of  the  Carnegie  Technical  Schools,  for  valuable  criticism  and 
advice,  and  to  Professors  Leigh  and  Palmer  for  the  answers  to 
many  of  the  problems.  The  imperfections  of  the  book  are, 
however,  the  author's  alone. 

For  the  drawing  of.  most  of  the  figures  the  author  is  indebted 
to  Mr.  John  R.  Boyd,  and  for  the  remainder  to  Mr.  Edwin  0. 
Kaul,   students   in  the   School  of  Applied  Science,  Carnegie 

Technical  Schools. 

N.  C.  RIGGS. 
Carnegie  Technical  Schools, 
August,  1910. 


CONTENTS 

CHAPTER  I 

Articles  1-18 

Graphical  Representation  op  Numbers.      Systems  op 
Coordinates 


PAGE 


Point  and  number — Addition  and  subtraction  of  segments  —  Coordi- 
nates of  points  in  the  plane  —  Cartesian  coordinates  —  Rectangu- 
lar coordinates  —  Polar  coordinates  —  Trigonometric  functions 
—  Relation  between  rectangular  and  polar  coordinates      .        .        1 

CHAPTER   II 

Articles  19-38 

Projections.     Lengths  and  Slopes  of  Lines.     Areas  op 
Polygons 

Projections  by  parallel  lines  —  Orthogonal  projection  —  Projection 
on  coordinate  axes  —  Distance  between  two  points  —  Angle 
between  two  lines  —  Inclination  and  slope  of  a  line  —  Ratio 
•into  which  a  point  divides  a  line  —  Angle  between  two  lines  of 
given  slopes  —  Condition  for  parallel  lines,  for  perpendicular 
lines  —  Area  of  a  triangle  —  Area  of  a  polygon  ....      15 

CHAPTER  III 

Articles  39-45 

Graphical   Representation  of  a  Function.      Equation  op 
A  Locus 

Function  and  variable  —  The  graph  of  a  function  —  Equation  of  a 

locus  —  Locus  satisfying  given  conditions  —  Intercepts       .        .      38 


viii  CONTENTS 

CHAPTER  IV 

Articles  46-50 

Locus  OF  AN  Equation 

PAGE 

Examples  —  Symmetry  —  Discussion  of  an  equation         ...      61 

CHAPTER  V 

Articles  51-54 

Transformation  of  Coordinates 

Translation  of  axes — Rotation  of  axes  —  Applications     ...      64 

CHAPTER  VI 

Articles  55-71 
The  Straight  Line 

The  equation  of  first  degree  —  Conditions  determining  a  straight 
line  —  The  two-point  equation — The  intercept  equation  —  The 
point-slope  equation  —  The  slope  equation  —  The  normal  equa- 
tion —  Reduction  of  Ax  +  By  -^  C  =  0  to  standard  forms  — 
Intersection  of  lines  —  Change  of  sign  of  Ax  +  By  -\-C  —  Dis- 
tance from  a  point  to  a  line  —  Equations  of  the  straight  line  in 
polar  coordinates 70 

CHAPTER   VII 

Articles  72-96 
Standard  Equations  of  Second  Degree 

The  circle  —  The  equation  x"^  +  y"^  +  Dx  +  Ey  +  F  =  0  —  Circle 
through  three  points  —  The  parabola  —  Axis  and  vertex  of  the 
parabola  —  Parameter  of  the  parabola  —  The  equation  y  =  ax^ 
-\-bx-\-  c  —  The  parabolic  arch  —  The  ellipse  —  Axes,  vertices, 
center  of  ellipse  —  The  hyperbola  —  Asymptotes  —  Axes,  cen- 
ter, vertices  of  hyperbola  —  The  conjugate  hyperbola  —  The 
equilateral  hyperbola  —  Reduction  of  the  general  equation  of 
second  degree  in  two  variables 88 


CONTENTS  ix 

CHAPTER   VIII 

Articles  97-107 

Trigonometric  and  Exponential  Functions 

PA6K 

The  sine  curve  —  Periodic  functions  —  The  exponential  curve  —  The 

logarithmic  curve  —  Graphs  —  Plotting  in  polar  coordinates       .     117 

CHAPTER   IX 

Articles  108-120 

Parametric  Equations  of  Loci 

Parametric  equations  of  circle  and  ellipse  —  Construction  of  the 
ellipse  —  The  cycloid  —  The  hypocycloid  —  The  epicycloid  — 
The  cardioid  —  The  involute  of  the  circle 129 

CHAPTER   X 

Articles  121-124 

Intersections  of  Curves.     Slope  Equations  of  Tangents 

Intersections  of  curves  —  Graphical  solutions  of  equations  —  Slope 

equations  of  tangents 142 

CHAPTER   XI 

Articles  125-154 

Slopes.     Tangents  and  Normals.     Derivatives 

Increments  —  Slope  of  a  curve  at  any  point  —  Equation  of  tangent 
to  a  curve  —  The  normal  —  Derivatives  —  Formulas  of  differen- 
tiation—  Geometric  meaning  of  the  derivative  —  Continuity  — 
Limit  of  ratio  of  a  circular  arc  to  its  chord  —  Radian  measure  of 
an  angle  —  Derivatives  of  the  trigonometric  functions        .        .     153 

CHAPTER   XII 

Articles  155-163 
Maxima  and  Minima.     Derivative  Curves 

Maximum  and  minimum  points  of  a  curve — Determination  of 
maxima  and  minima — The  first  derivative  curve  —  Concavity 
—  The  second  derivative  curve 178 


X  CONTENTS 

CHAPTER  XIII 

Articles  164-177 
The  Conic  Sections 

PAGB 

Construction  of  conies  —  Vertices  —  Classification  of  conies  —  Equa- 
tion of  the  conic  in  rectangular  coordinates  —  The  parabola  — 
The  centric  conies  —  The  ellipse  —  The  hyperbola  —  Axes —  ' 
The  equation  of  the  conic  in  polar  coordinates   .        .        .        .192 

CHAPTER   XIV 

Articles  178-186 

Pkopekties  of  Conics 

Subtangent  and  subnormal  of  the  parabola  —  Property  of  reflection 
of  the  parabola  —  Focal  radii  of  ellipse  and  hyperbola  —  Prop- 
erty of  reflection  of  ellipse  and  hyperbola 205 

CHAPTER   XV 

Articles  187-195 

The  General  Equation  of  Second  Degree 

Removal  of  the  terms  of  first  degree  —  Removal  of  the  term  in  xy  — 

Locus  of  the  equation 214 

CHAPTER   XVI 

Articles  196-201 

Empirical  Equations 

Points  lying  on  a  straight  line  —  The  curve  y  ■=  cx"^  —  The  curve  y  = 
ah'  —  Some  special  substitutions  —  The  curve  y  =  a  +  bx  -^  cx^ 
+  "'  +kx^ 223 

CHAPTER   XVII 

Articles  202-212 

COOEDINATES    IN   SpACB 

Rectangular  coordinates  in  space  —  Distance  between  two  points  — 
Polar  coordinates  —  Relation  between  rectangular  and  polar 
coordinates — Relation  between  the  direction  cosines  of  a  line 
—  Direction  cosines  of  a  line  joining  two  points  —  Spherical 
coordinates  —  Projection  of  a  broken  line  —  Angle  between  two 
lines 233 


CONTENTS  xi 

CHAPTER  XVIII 

Articles  213-217 
Loci  and  their  Equations 

PAGE 

Certain  lines  and  planes  —  Cylinders  —  Surfaces  of  revolution  — 
Plane  sections  of  a  locus  —  Locus  of  an  equation  in  three 
variables 243 

CHAPTER   XIX 

Articles  218-225 

The  Plane  and  the  Straight  Line 

The  normal  equation  of  the  plane  —  The  intercept  equation  of  the 
plane  —  General  equation  of  first  degree  in  three  variables  — 
Distance  from  a  point  to  a  plane  —  Angle  between  two  planes  — 
Equations  of  the  straight  line 249 

CHAPTER   XX 

Articles  226-234 

The  Quadric  Surfaces 

The  ellipsoid  —  Hyperboloid  of  one  sheet  —  Hyperboloid  of  two 
sheets  —  Elliptic  paraboloid  —  Hyperbolic  paraboloid  —  The 
cone  —  The  conic  sections 257 

CHAPTER   XXI 

Articles  235-239 

Space  Curves 

The  helix  —  Curve  of  intersection  of  two  cylinders  —  Curve  of  inter- 
section of  cylinder  and  sphere  —  General  equations  of  a  space 
curve 268 

CHAPTER   XXII 

Articles  240-246 

Tangent  Lines  and  Planes 

Partial  derivatives  — Tangent  plane  to  a  surface  —  Normal  to  a  sur- 
face —  Tangent  line  to  a  space  curve 272 


ANALYTIC  GEOMETRY 


CHAPTER   I 

GRAPHICAL  REPRESENTATION  OF   NUMBERS.    SYSTEMS 
OF   COORDINATES 

I.   POINTS  ON   A  STRAIGHT  LINE 

1.  Point  and  Number.  On  a  straight  line  let  a  fixed  point 
0  be  taken  from  which  to  measure  distances,  and  let  a  definite 
length  be  chosen  as  a  unit.  If  this  unit  be  laid  off  in  succession 
on  the  line,  beginning  at  0,  other  points  of  the  line  are  obtained 
whose  distances  from  0  are  1,  2,  3,  •••,  etc.  times  the  unit  dis- 
tance. It  is  convenient  to  think  of  these  points  as  represent- 
ing the  numbers,  or  of  the  numbers  as  representing  the  points. 

Thus  a  point  7  units  from  0  may  be  taken  to  represent  the 
number  7,  and  conversely  the  number  7  may  be  said  to  repre- 
sent the  point. 

Since  there  are  two  points  of  the  line  at  the  same  distance 

from  P,  one  to  the  right,  the        „  ^  ,.       ,. 

P  0  P,       P 

other  to  the  left,  and  since  ^ 


there  are   both  positive  and       -4-3-2-1     0     1     2     3    4 
negative   numbers,  let  it   be  ^^^"  ^' 

agreed  that  points  to  the  right  of  0  shall  represent  positive 
numbers  and  those  to  the  left  of  0  negative  numbers. 

Thus  a  point  3  units  to  the  right  of  0  represents  the  number 
3,  and  a  point  3  units  to  the  left  of  0  represents  the  number 
—  3.     The  numbers  are  also  said  to  represent  the  points. 

It  can  be  shown  that  to  every  point  of  the  line  there  corre- 
sponds a  real  number,  and  conversely,  to  every  real  number 
there  corresponds  a  point  of  the  line.  The  whole  system  of 
real  numbers  may  therefore  be  represented  by  points  on  a 

B  1 


2  ANALYTIC  GEOMETRY 

straight  line  with  one  number  for  each  point  and  one  point  for 
each  number. 

The  point  0  is  called  the  origin.  It  represents  the  number 
zero. 

2.  Notation.  If  P  is  any  point  of  the  line  and  0  is  the  ori- 
gin, the  symbol  OP  is  used  to  denote  the  number  which  repre- 
sents the  point  P. 

E.g.  if  P  lies  3  units  to  the  right  of  0,  then  OP  is  3 ;  while 
if  P  lies  3  units  to  the  left  of  0,  OP  is  -  3. 

It  is  convenient  to  denote  the  number  which  represents  a 
point  by  a  single  letter,  as  x;  thus  OP=x.  Then  if  P  lies  to 
the  right  of  0,  a;  is  a  positive  number,  and  if  P  lies  to  the  left 
of  0,  cc  is  a  negative  number. 

Different  points  on  the  line  will  sometimes  be  denoted  by  P 
with  different  subscripts,  and  the  numbers  representing  these 
points  by  x  with  corresponding  subscripts. 

Thus,  in  Fig.  1,     OP^  =  x^  =  2,     OP^  =  a^a  =  -  4. 

3.  Segments  of  the  line.  In  speaking  of  any  segment  of  the 
line,  as  AB,  the  first  letter  named  is  called  the  beginning,  and 
the  last  letter  the  end,  of  the  segment. 

Thus  A  is  the  beginning,  and  B  is  the  end,  of  AB,  while  B  is 
the  beginning,  and  A  is  the  end,  of  BA. 

It  is  important  to  represent  the  value  of  any  segment  of  the 
line  by  a  number,  and  this  is  done  by  defining  the  value  of 
any  segment  of  the  line  to  be  the  number  which  would  repre- 
sent the  end  of  the  segment  if  the  beginning  of  the  segment 
were  taken  as  origin. 

Thus,  in  Fig.  2,  with  0  as  origin, 

P,    P3  A  0  P2         B 

.    I      I      I 


^5  =  6,  5^  =  - 6,^30  =  5,  0Pi  =  -6,  0^=-2,  P3Pi  =  -l, 
P,0  =  -2. 


GRAPHICAL  REPRESENTATION  OF  NUMBERS        3 

From  the  definition  of  the  value  of  a  segment  it  follows  that 
the  value  of  any  segment  read  from  right  to  left  is  negative, 
while  the  value  of  any  segment  read  from'  left  to  right  is 
positive. 

EXERCISE  I 

1.  What  numbers  represent  the  points  Pi,  P2,  P3,  -4,  B,  in  Fig.  2  ? 

2.  What  are  the  values  of  P2P3,  P1P2,  P3P2,  BP3,  P3B  ? 

3.  If  A  be  taken  as  origin,  what  are  the  numbers  that  represent  Pi,  P2, 
O,  P3,  P? 

4.  If  the  origin  be  moved  two  units  to  the  right,  how  are  the  numbers 
representing  different  points  affected  ?  How  if  the  origin  be  moved  h 
units  to  the  right  ?  to  the  left  ? 

4.  Change  of  sign  of  a  segment.  Since  any  segment  AB  of 
the  line  contains  the  same  number  of  units  as  BA,  but  is  meas- 
ured in  the  opposite  direction,  it  follows  that 

BA  =  -AB,  or  BA  +  AB  =  0. 

5.  Addition  of  segments.  Let  A,  B,  and  C  be  any  three 
points  on  the  line.     Then 

AC  =  AB  +  BC. 

Proof.     Three  cases  arise : 

(1)  B  between  A  and  G,  (2)  A  between  B  and  C,  (3)  0  be- 
tween ^  and  B. 

(^)   A        B         C  C        B 

(2) 

(3) 

Fig.  3. 

In(l),  AO=AB-\-BO', 

in  (2),  AC=BC-BA  =  BC-hAB,hj  Art.  4, 

or  AC=:AB-\-BC; 

in  (3),  AC  =  AB-CB  =  AB-\-BO, 


A        B 

C 

>    B          A 

c 

1^     A     C 

B 

C       A 

B 

B      C 

A 

4  ANALYTIC  GEOMETRY 

6.  Subtraction  of  segments.  By  writing  —  CB  instead  of  BC 
in  the  equation  of  Art.  5,  namely, 

AC=AB-^BC, 

that  equation  becomes 

AC  =  AB-CB. 

The  results  found  in  this  and  the  preceding  articles  lead  to  the 
rules  for  geometric  addition  and  subtraction  of  numbers  that 
follow. 

7.  Geometric  addition  of  numbers.     Let  P^  and  P^  be  two 

points  on  the  line  represented  by  the  numbers  x^  and  X2  respec- 
tively.    Then  0 A  =  x^,  OP^  =  x^. 

Three  cases  arise : 

(1)  both  numbers  positive ;  (2)  one  number,  say  x-^,  negative, 

the  other  positive ;  (3)  both 

0)    — I h! h^ 1 numbers  negative. 

rg)  Pi       O P        Pg  To     represent     geometri- 

p  p    p  Q     cally  the  sum  of  x^  and  x^  lay 

^^^   ^ 1 — I •—  off   from   the   end   of   x^   sl 

^^^-  *•  segment,   PiP,   equal   to  X2 

and  measured  from  Pj  in  the  same  direction  as  x^  is  measured 
from  0.     Then 

0P=Xi-{-X2. 

For,  in  each  case,  0P=  OP^  +  P^P,  by  Art.  5, 

=  0P,^0P2  M 

—  37j  ~}~  fl/g. 

8.  Geometric  subtraction  of  numbers.  Consider  again  the 
three  cases  of  Art.  7.  To  represent  geometrically  the  differ- 
ence Xi  —  X2  lay  off  from  the  end  of  Xi  a  segment  PiP  equal  to 
—  X2,  i.e.  having  the  same  numerical  value  as  X2  but  opposite 
in  direction. 


P  0        Pi  P, 

i 1 1 1 

P  Pi    0  P2 

H 1 1 1 

Pi  P       P2  o 


GRAPHICAL  REPRESENTATION  OF  NUMBERS        5 

Then  0P  =  Xi-X2. 

For,  in  each  case, 

0P=  OP,  +  F,P=  0P-PP,=  0P^-0P,  =  x,-X2. 

Another,  and  more  important,  expression  of  the  difference  is 
as  follows : 

x^  -  X,  =  OP,  -  OP2  =  OP,  -\-PM 
=  P,0-{-0P,=  P2Pi,  by  Art.  5, 
or  JP1-P2  =  a52  —  a?i. 

Hence,  the  value   of  any  seg- 
ment of  the  line  is  equal  to  the 

number  that  represents  the  end  minus  the  number  that  represents 
the  beginning  of  the  segment. 

This  principle  will  be  of  frequent  use  hereafter. 

Illustration.     In  Fig.  6,  if  P„  P.>,  P3  are  three  points  on 

P2    .      P4    0     .      Pi  Pa 

- — t 1 1 1 (— — ^! — I 1^ — 

-  3  -  f     Fig.  6.     ^  ^ 

the  line  represented  by  the  numbers  2,   —  3,  4  respectively, 
then 

PiP2  =  -3-2  =  -5,  P,Pi  =  2-(-3)  =  5, 
P3P,  =  -3-4  =  -7,  PiP3  =  4-2  =  2,  P3Pi  =  2-4  =  -2. 

9.  Relative  position  of  points  representing  numbers.  Let  x^ 
and  X2  be  any  two  real  numbers  represented  by  the  points  P, 
and  Pa  respectively. 

By  Art.  8,  -P2A  =  x,—  x^. 

Now  if  Xi  >  072  then  x,  —  x^  is  positive,  and  conversely. 

Therefore,  if  x^^x^,  P^P,  is  positive,  and  hence  Pi  lies  to 
the  right  of  Pg ;  if  a^i  <  %,  AA  is  negative,  and  hence  Pi  lies 
to  the  left  of  Pg,  and  conversely. 

Hence,  of  the  two  points  which  represeyit  ttvo  real  numbers  the 
point  which  represents  the  greater  number  lies  farther  to  the  right. 

E.g.  in  Fig.  6,  P,,  which  represents  2,  lies  to  the  right  of  Pg, 
which  represents  —3;  P4,  which  represents  —1,  lies  to  the 


6  ANALYTIC  GEOMETRY 

right  of  P^j  which  represents  —  3.  This  agrees  with  the  state- 
ment that  2  is  greater  than  —  3,  and  that  —  1  is  greater  than 
-3. 

EXERCISE   II 

1.  Represent  geometrically  the  following  pairs  of  numbers,  their  sum, 
the  first  minus  the  second,  the  second  minus  the  first : 

(a)  3,  2.         (6)  -  2,  3.         (c)  .4,  -  3.         (d)  -  5,  -  1. 

2.  In  Fig.  5  express  the  following  segments  as  the  difference  of  the 
numbers  representing  the  points:  P1P2,  P2-P1,  P2O,  OPi,  PiO,  OP2. 

vvk^  3.  In  Fig.  5  what  segments  represent  Xx  —  iC2,  x^  —  Xx^  iCi,  X2,  —  Xi^ 

4.    In  Fig.  6,  by  means  of  the  principle  in  Art.  8,  find  the  values  of 
P2P1,  P3P1,  P4O,  P4P3,  P3P4,  OP3,  P3O,  P3P2. 


n.  COORDINATES  OF   POINTS  IN  THE   PLANE 

10.  Location  of  a  point.  To  determine  the  position  of  a 
point  on  a  straight  line  one  magnitude  is  sufficient;  namely, 
the  distance  of  the  point,  right  or  left,  from  a  fixed  point  of 
the  line.  The  number  that  represents  a  point  on  the  line 
determines  the  position  of  the  point  when  the  origin  is  given. 

In  the  plane,  however,  two  magnitudes  are  necessary  to 
determine  the  position  of  a  point. 

There  are  many  ways  of  choosing  these  magnitudes.  Two 
simple  methods,  and  the  only  ones  used  in  this  book,  are  to 
consider  the  location  of  the  point,  (1)  with  reference  to  tw^ 
intersecting  straight  lines,  (2)  with  reference  to  a  fixed  Lnv" 
and  a  fixed  point.  A  consideration  of  these  two  methods 
leads  to  the  definitions  of  (1)  Cartesian  Coordinates,  (2)  Polar 
Coordinates. 

11.  Cartesian  coordinates.  Let  two  intersecting  straight 
lines,  OX  and  OF,  be  taken  as  lines  of  reference  and  an  arbi- 


Fig.  7. 


GRAPHICAL  REPRESENTATION  OF  NUMBERS        7 

trary  length  be  chosen  as  a  unit.  Then  to  every  point  P  in  the 
plane  there  can  be  assigned  a  pair  of  real  numbers  as  follows : 
Through  the  point  P  draw  lines  parallel  to  OX  and  0  Y,  meet- 
ing OX  and  OY  in  M  and  JST 
respectively.  The  pair  of  num- 
bers which  measure  JSfP  and 
MP  is  taken  to  represent  the 
point  P.  To  every  position  of 
P  there  corresponds  one,  and 
only  one,  pair  of  such  num- 
bers. In  order  that  to  every 
pair  of  real  numbers  there  may 
correspond  one,  and  only  one, 
point,  some  agreement  in  re- 
gard to  signs  is  necessary.  To 
the  agreement  already  made  that  a  segment  measured  from  left 
to  right  shall  be  positive,  and  one  measured  from  right  to  left 
shall  be  negative,  let  there  be  added  the  agreement  that  a  seg- 
ment measured  upward  shall  be  positive,  and  a  segment  meas- 
ured downward  shall  be  negative.  With  this  agreement  in 
regard  to  signs  there  corresponds  one,  and  only  one,  point  in 
the  plane  to  every  pair  of  real  numbers. 

The  lines  OX  and  OY  are  called  the  a?-axis  and  7/-axis 
repectively. 

The  segments  NP  and  MP  are  called  respectively  the 
abscissa  and  ordinate  of  P,  and  together  are  known  as  the 
Cartesian  coordinates  of  the  point. 

It  should  be  carefully  noted  that,  from  the  definition,  the 
abscissa  of  P  is  measured  from  the  y-axis  to  P,  and  the  ordi- 
nate of  P  is  measured  from  the  x-axis  to  P. 

The  abscissa  and  ordinate  are  most  frequently  denoted  by  x 
and  y  respectively,  though  other  letters  are  sometimes  used. 

The  point  P  is  denoted  by  the  coordinates  inclosed  in 
parentheses  and  separated  by  a  comma,  thus,  (x,  y)  or 
P(x,   y). 


8 


P2(«2.2/2) 


ANALYTIC  GEOMETRY 

Y/  To  distinguish  one  point 

from     another,     subscripts 
are  often    used.      Thus   in 
P^i^^^yO  Fig.  8, 

X2  =  JSr2P2=03Io  =  -2, 
y,  =  M,P,=  0^2  =  3, 

x^  =  N^Ps  =  —  3, 


'4  (2:4 » 2/4) 


Fig.  8. 


ys  =  MsPs  =  -S,  etc. 


M,M,  =  M,0+OM,  =  -x,  +  x„ 
M^Mi  =  M^O  +  03fi  =  -x^-\-  xi,  etc. 


EXERCISE  III 

1.  Assume  a  pair  of  axes  and  locate  the  points  (2,  3),    (2,   —3), 
(-2,  4),   (-5,   -6),   (0,  2),   (4,  0),   (-1,  0),   (0,  -3),  (0,  0). 

2.  In  Fig.  8  express  as  the  dif- 
ference of  two  abscissas,  3f2Ms, 
M^Mi,  MsM^,  MoMi. 

3.  Express  as  the  difference 
of  two  ordinates,  N2N3,  iViiV4, 

4.  What  segments  represent 
Xi  —  X2,  X3  -  Xi,  Xi  —  Xi,  X3  -  res  ? 

5.  What  segments  represent 
y2  -  Vu  Vi  -  «/2,  ys  -  2/2, 2/1  -  y*  ? 

6.  Where  do  all  points  lie  that 
have  the  abscissa  zero  ;  that  have 
the  ordinate  zero  ? 

7.  Where  do  all  points  lie  that 
have  the  abscissa  2  ;    that  have  the  abscissa  —  3 ;  that  have  the  ordinate 
2  ;  that  have  the  ordinate  —  4  ? 


Fig.  9. 


GRAPHICAL  REPRESENTATION  OF  NUMBERS        9 

8.  In  Fig.  9  express  as  the  difference  of  two  abscissas,  PiB,  P^^-,  PzQi 
IB,  SN;  and  as  the  difference  of  two  ordinates,  TPs,  PnV,  BQ,  TS, 
P2B,  PaS. 

9.  In  Fig.  9  let  Pi,  Pg,  Ps  have  coordinates  (2,  3),  (4,  —  2),  and 
(-3,  2),  respectively,  and  find  the  values  of  SN,  P2S,  BQ,  BT,  PiN, 
SPs,  P2P,  QP3. 

12.  Segments  not  parallel  to  an  axis.  Segments  of  lines  not 
parallel  to  one  of  the  coordinate  axes  will  not  have  definite 
signs  given  to  them.  They  vi^ill  generally  be  considered  as 
positive  lengths,  but  where  the  two  opposite  directions  along 
the  same  straight  line  are  considered,  one  of  them  will  be 
counted  as  opposite  in  sign  to  the  other. 

13.  Rectangular  coordinates.  If  the  axes  in  the  Cartesian 
coordinate  system  are  at  right  angles  to  each  other,  the  system 
is  called  the  rectangular  system  of  coordinates. 

This  system  possesses  the  advantage  of  simplicity,  in  many 
problems,  over  that  of  oblique  axes,  and  as  most  of  the  proper- 
ties and  relations  of  figures  to  be  studied  do  not  depend  upon 
the  system  of  coordinates  used,  the  rectangular  system  will  be 
used  except  where  otherwise  indicated. 

14.  Polar  coordinates.  The  position  of  a  point  in  the  plane 
may  be  determined  by  the  length  of  the  line  joining  it  to  a  fixed 
point,  and  the  angle  which  this  line 
makes  with  a  fixed  direction. 

In  Fig.  10  let  0  be  a  fixed  point 
and  OA  a  fixed  line.  Let  P  be  any 
point  in  the  plane.  Then  the  seg- 
ment OP  and  the  angle  AOF  deter- 
mine the  location  of  P.  Fig.  10. 

The  segment  OP  is  called  the  radius  vector,  and  the  angle 
AOP  the  vectorial  angle  of  P. 

Together  they  are  known  as  the  polar  coordinates  of  P.  They 
are  usually  denoted  by  r  and  6,  respectively,  and  the  point 
indicated  by  (i-,  6),  or  P(r,  6). 


10 


ANALYTIC  GEOMETRY 


The  fixed  point  0  is  called  the  origin,  or  pole ;  the  fixed  line 
OA  the  initial  line,  or  axis. 
The  line  OP  is  called  the  terminal  line  of  the  angle  AOP. 
With  these  definitions  it  is  easy  to  see  that  any  point  in  the 
plane  may  be  represented  by  polar  coordinates,  both  of  which 
are  positive,  and  with  the  angle  less  than  360°. 

In  order,  however,  to  represent  both  positive  and  negative 
numbers  by  points,  the  following  agreement  in  regard  to  signs 

is  made:  Positive  angles 
will  be  measured  in  the 
counter-clockwise  direction 
from  the  initial  line ;  nega- 
tive angles  in  the  opposite 
direction.  By  a  negative 
radius  vector  will  be  meant 
one  laid  off  on  the  terminal 
line  of  the  vectorial  angle 
produced  back  through  the 
pole. 

3 


27r 

3 


and 


5, 


would  be  as  indi- 


Thus,  the  points  (5, 

Gated  in  Fig.  11. 

With  the  above  agreement  in  regard  to  signs  it  follows  that 
to  every  pair  of  coordinates 
there  is  just  one  point  in  the 
plane,  but  to  every  point  in 
the  plane  there  corresponds  an 
indefinite  number  of  pairs  of 
coordinates. 


J^.gr.  thepoint  [2,  -  J  may  also 
be  represented  by  [  —  2, 


(-2,*f)(2,-f)     or     b. 


GRAPHICAL  REPRESENTATION  OF  NUMBERS      11 


any  other  pair  of  coordinates  obtained  by  increasing  the  angle 
of  any  of  the  above  pairs  by  an  integral  multiple  of  2  tt,  the 
radius  vector  being  unchanged. 

If  d  is  restricted  to  being  numerically  less  than  2  tt,  the  four 
pairs  of  values  written  above  are  the  only  ones  that  represent 
the  given  point. 

Note.  The  student  should  remember  that  the  unit  of  circular  measure 
of  an  angle  is  the  angle  subtended  at  the  center  of  a  circle  by  an  arc  equal 
in  length  to  a  radius  of  the  circle.     This  unit  is  called  the  radian. 

From  the  definition  it  follows  that  tt  radians  =  180°,  where  tt  =3.14159  ••  •. 

When  an  angle  is  represented  by  a  letter  or  figure  without 
the  degree  sign  (°),  it  will  be  understood  that  the  unit  of  meas- 
ure is  the  radian. 

EXERCISE  IV 

1.  Plot  in  polar  coordinates  (2,  -30^),  f-4,  ^V  ll,  -^), 
(tt,  tt),  (tt,  7r°),  (3,  2). 

2.  Plot  in  rectangular  coordinates  (—3,  4),  (0,  —3),  (0,  0),  (a,  0), 
(0,.),(.,-2f),(6,|). 


HI.    THE   TRIGONOMETRIC   FUNCTIONS 

15.  Definitions    of    the    trigonometric   functions. 

given  any  angle,    P' 

assume  a  system 

of  rectangular 

coordinates     and 

place   the  vertex 

of   the    angle   at 

the    origin,    with 

the    initial    line 

coinciding     with 

the  positive  part 

of     the     a?- axis ; 

,       ,  '  Fig.  13. 

positive  angles  to 

be  reckoned  counter-clockwise  and  negative  angles,  clockwise. 


12  ANALYTIC  GEOMETRY 

Assume  any  point  P  on  the  terminal  line ;  let  its  coordinates 
be  X  and  y,  and  its  distance  from  the  origin  be  r,  counted 
always  positive.  Then,  whatever  the  size  of  the  angle,  the 
following  definitions  are  given : 

sine  of  ^  =  ordinate/distance  =  y/r^ 

cosine  of  ^  =  abscissa/distance  =  x/r, 

tangent  of  A  =  ordinate/abscissa  =  y/Xj 

cotangent  of  ^  =  abscissa/ordinate  =  x/y, 

secant  oi  A  =  distance/abscissa  =  r/x, 

cosecant  of  ^  =  distance/ordinate  =  r/y. 

16.  Formulas  and  tables.  A  set  of  the  more  important 
formulas  connecting  the  trigonometric  functions  of  angles, 
and  a  table  of  sines,  cosines,  and  tangents  are  given  at  the 
back  of  the  book. 

17.  The  inverse  trigonometric  functions.  The  symbol  sin*' £c, 
read  "  anti-sine  a?,"  is  used  as  equivalent  to  the  words,  "an 
angle  whose  sine  is  ic." 

Thus  one  value  of  sin~^  (i)  is  - ,  or  30°;  another  value  is  — . 

6  6 

In  like  manner  the  symbols  cos~^  x,  tan~^  x,  etc.,  are  used  as 

equivalent  to  "  an  angle  whose  cosine  is  a;,"  "  an  angle  whose 

tangent  is  ic,"  etc. 

EXERCISE  V 

1.  Find  by  the  use  of  the  table  the  sine,  cosine,  and  tangent  of  each  of 
the  following  angles  :    20^,  17*^  20',  185°,  109°  40',  290°,  165^  .2  radian, 

.72  radian,  (-J  radian. 

2.  Given  A  =  sin-i  .6,  find  a  value  of  A  in  the  first  quadrant,  and  one 
in  the  second  quadrant. 

3.  Given  A  =  tan-i  .4563,  find  two  values  of  A. 

4.  Find  sin-i(tan25°},  sin(tan-i3.26),  sin(sin-i  ,35). 

5.  Show  that  sin  (sin- 1 .5)  =  .5,  and  that  sin-i(sin30°)=  30°,  or  150'', 
or  390°,  etc. 


GRAPHICAL  REPRESENTATION  OF  NUMBERS      13 


18.   Relation  between  rectangular    and   polar    coordinates. 

Let  the  origin  in  the  two  systems  be  the  same,  and  let  the 

initial  line  coincide  with 

the  positive  part   of  the  pc^,  y\ 

ic-axis. 

Let  P  be  any  point  in 
the  plane  with  rectangular 
coordinates  x  and  y  and 
polar  coordinates  r  and  0 
(Fig.  14),  the  polar  coordi- 
nates being  so  chosen  that 
e=Z.  XOP  and  r  =  OP, 
where  OP  is  positive. 


Fig.  14. 
Then  from  the  definition  of  sine  and  cosine. 


-  =  cos  ^,  ^  =  sin  e, 
r  r 


or 


(1) 


Qc  z=r  cos  6, 
y  =r  sin  6. 

These  equations  express  x  and  y  in  terms  of  r  and  0.  From 
the  figure,  or  from  these  equations,  r  and  6  can  be  expressed 
in  terms  of  x  and  y.     The  resulting  equations  are 


r  =  v^+  y% 
9  =  tan-^  (^y 


(2) 


EXERCISE   VI 

1.  Show  how  to  obtain  eqs.  (2)  of  Art.  (18)  from  eqs.  (1). 

2.  Show  that  if  the  polar  coordinates  of  F  be  chosen  so  that  d  differs 
from  Z  XOP  by  180°,  and  r  is  the  negative  of  OP,  eqs.  (1)  still  hold. 

3.  Find  the  polar  coordinates  of  the  points  whose  rectangular  coor- 
dinates are  (3,  -  7),  (4,  3),  (-  2,  1),  (-  4,  -  2). 

4.  Find  the  rectangular  coordinates  of  the  points  whose  polar  coor- 
dinates are  (2,  30'^),  (-  3,  45°),  (4,  -  60"),  (-  2,  -  15°). 

5.  In  rectangular  coordinates  where  do  all  points  lie  whose  abscissas 
are  zero ;  whose  ordinates  are  zero  ;  whose  abscissas  equal  any  constant 


14  -       ANALYTIC  GEOMETRY 

C ;  whose  abscissas  equal  their  ordinates  ;  whose  abscissas  equal  the  nega- 
tive of  their  ordinates  ? 

6.  What  is  true  of  the  polar  coordinates  of  points  which  satisfy  each 
of  the  conditions  of  example  5  ? 

7.  In  polar  coordinates  where  do  all  points  lie  whose  vectorial  angles 
are  zero  ;  whose  vectorial  angles  equal  30*^ ;  whose  vectorial  angles  equal 
any  constant ,-  whose  radii  vectores  equal  5 ;  whose  radii  vectores  equal 
any  constant  C  ? 

8.  What  equation  is  true  of  the  rectangular  coordinates  of  the  points 
which  satisfy  each  of  the  conditions  in  example  7  ? 

9.  Find  the  polar  coordinates  of  the  point  whose  rectangular  coor- 
dinates are  (3.26,  -2.67). 

10.   Find  the  rectangular  coordinates  of  the  point  whose  polar  coor- 
dinates are  (6.34,  34°  16'). 


CHAPTER   II 

PROJECTIONS.    LENGTHS  AND  SLOPES  OF  LINES.    AREAS 
OF  POLYGONS 

L   PROJECTIONS 

19.  Projections  by  parallel  lines.  Through  the  beginning 
and  end  of  a  segment  AB  let  lines  parallel  to  a  given  direc- 
tion be  drawn  to  intersect  a  given  line  MN  in  C  and  D  respec- 
tively. Then  CD  is  called  the  projection  of  AB  on  MJ^,  for 
the  given  direction. 

The  beginning  and  end  of  the  projection  are  to  be  read  in 
the  same  order  as  the  beginning  and  end  of  the  segment. 


Thus  CD  is  the  projection  of  AB,  while  DC  is  the  projection 
of  BA.     (Fig.  15.) 

The  direction,  parallel  to  which  the  lines  AC  and  BD  are 
drawn,  is  called  the  direction  of  projection. 

Evidently,  the  value  of  the  projection  depends  upon,  (1)  the 
length  of  the  segment,  (2)  the  difference  in  direction  of  the 
segment  and  the  line  on  which  it  is  projected,  and  (3)  upon 
the  direction  of  projection.     It  is  evident,  also,  that  the  pro- 

15 


16 


ANALYTIC  GEOMETRY 


jections  of  a  given  segment  on  parallel  lines  are  equal,  if  the 
direction  of  projection  is  the  same. 

20.  Orthogonal  projection.  If  the  direction  of  projection  is 
perpendicular  to  the  line  on  which  the  segment  is  projected, 
the  projection  is  called  orthogonal. 

B 


Fig.  16. 

Thus  in  Fig.  16   CD  is  the  orthogonal  projection  of  AB 

on  MN. 


21.  Projection  in  the  direction  of  one  coordinate  axis  on  a  line 
parallel  to  the  other  axis. 

Definitiox.  The  projection  in  the  direction  of  the  2/-axis 
of  a  segment  on  a  line  parallel  to  the  a;-axis  will  be  called  the 
ic-projection  of  the  segment. 

A  similar  definition  is  given  for  the  y-projection  of  the  seg- 
ment. 

Consider  now  the  ic-projection  of  any  segment  P^Pi. 

Let  the  coordinates  of  Pj  and  P.,  be  {x'^,  y^  and  (x^,  y^  re- 
spectively. Three  cases  may  arise  :  P^Pi  may  lie  wholly  to  the 
right  of  the  ly-axis,  may  cut  the  ?/-axis,  or  may  lie  wholly  to  the 
left  of  the  2/-axis.     (Fig.  17.) 

Let  the  projection  in  either  case  be  M^M,,  and  let  the  line  on 
which  P1P2  is  projected  meet  the  ?/-axis  at  N,  Then,  in  either 
case, 

M1M2  =  MiN+  NM2  =  —  Xi-\-x<>  =  X2  —  iCj. 


PROJECTIONS.     LENGTHS  AND  SLOPES  OF  LINES     17 

Therefore,  the  x-projection  of  a  segment  is  equal  to  the  abscissa 
of  the  end  of  the  segment  minus  the  abscissa  of  the  beginning. 

Yi 


In  like  manner  it  can  be  shown  that  the  y-projection  of  a  seg- 
ment is  equal  to  the  ordinate  of  the  end  minus  the  ordi7iate  of  the 
beginning. 

Example.  The  a>projection  of  the  segment  from  Pi(—  1,  3) 
to  P2(3,  2)  is  3  —  (—  1)  =  4,  and  the  ^/-projection  is  2  —  3  =  —  1. 

EXERCISE  VII 

1.  Prove  that  the  ^/-projection  of  a  segment  is  equal  to  the  ordinate  of 
the  end  of  the  segment  minus  the  ordinate  of  the  beginning. 

2.  Find  the  x-  and  ^/-projections  of  the  segments  from  the  first  to  the 
second  of  each  of  the  following  pairs  of  points:  (2,  3),  (—2,  6); 
(-3,  -1),  (4,  -5);  (1,  -2),  (3,7);  («,&),  (c,  d);  (0,  1),  (-2,  0); 
(0,0),  (3,  -5);   (u,v),  (s,t). 

Check  the  results  by  drawing  the  figure  in  each  case. 

3.  If  the  axes  are  at  right  angles  to  each  other,  find  the  distance  from 
the  origin  to  (3,  7) ;  from  the  origin  to  (x,  y). 

c 


18  ANALYTIC  GEOMETRY      • 

4.  If  the  axes  are  at  right  angles  to  each  other,  find  the  distance 
between  (-  5,  3)  and  (2,  -  6). 

5.  If  the  axes  are  rectangular,  show  that  the  distance  between  (iCi,  yi) 
and  (X2,  y-i)  is  V(a:i  -  x^y^  +  {yi  -  y2)\ 

6.  In  rectangular  coordinates  the  point  (x,  y)  moves  so  as  to  keep  at 
the  distance  5  from  the  origin.  Express  this  by  means  of  an  equation. 
What  is  the  locus  of  the  point  ? 

7.  What  is  the  a;-projection  of  a  segment  parallel  to  the  y-axis ;  the 
t/-projection  of  a  segment  parallel  to  the  x-axis  ? 

8.  The  vertices  of  a  triangle  are  A,  B,  and  G.  Show  that  the  sum  of 
the  projections  of  AB,  BC,  and  CA  on  any  line  is  zero,  and  that  the 
projection  of  AC  =  the  projection  of  AB  +  the  projection  of  BC. 

9.  Show  that  the  sum  of  the  projections  of  the  sides  of  any  closed 
polygon  taken  in  order,  i.  e.  so  that  the  beginning  of  each  side  is  the  end 
of  the  preceding,  on  any  line  is  zero. 

10.  Show  that  if  the  sum  of  the  projections  of  the  sides  of  a  polygon 
taken  in  order  on  one  straight  line  is  zero,  the  polygon  is  not  necessarily 
closed  ;  but  if  the  sum  of  the  projections  taken  in  order  on  two  non- 
parallel  lines  is  zero,  the  polygon  is  closed. 

II.    LENGTHS  AND   SLOPES  OF   SEGMENTS.    DIVISION 
OF   SEGMENTS 

22.   Distance  between  two  points.     Numerical  examples. 

Example  1.  To  find  the  distance  between  the  two  points 
whose  Cartesian  coordinates  are 
(2,  -4)  and  (-3,  5),  the  angle  be- 
tween the  axes  being  G0°. 

Let  (2,  -4)  be  P„  and  (-3,  5) 
be  Pa- 
Through   Pi    and  Pg   draw   lines 
parallel,  respectively,  to  the  x-  and 
?/-axes,  intersecting  in  Q.     (Fig.  18.) 
jj,j^  ^g  By  the  law  of  cosines  from  trigo- 

nometry, 

P^l=QPl+ QPl- 2  QPi- W,  cos  P^QP^, 


PROJECTIONS.     LENGTHS  AND  SLOPES  OF  LINES     19 
Here  QP^  =  2  -  (-  3)  =  5,  by  Art.  21, 

cosPiQP2  =  cos60°  =  i. 
.•.P,P2  =  V6i  =  7.81  ...  . 
Example  2.     To  find  the  distance  between  the  points  whose 

polar  coordinates  are  [2,  —  j  and  (5,  — '^ 


lPi 


Fig.  19. 

Let  (2,  ^\  be  P„  (5,  - 1")  be  P^ 

and  let  P^Po  =  d.     (Fig.  19.) 
By  trigonometry, 

d'  =  OPI  +  OPI  -  2  OPi  .  OPa  cos  Pi  OP2 

=  4  +  25-2. 2. 5cos^ 
6 

=  4  +  25  +  20  cos^ 
6 

=  29  +  17.32  ...  . 


,'.d  =  V46.32  =  6.81  nearly. 

EXERCISE  VIII 

1.  If  the  angle  between  the  axes  is  45°,  find  the  distance  between  the 
points  (-.3,  5)  and  (4,  1). 

2.  If  the  angle  between  the  axes  is  80°,  find  the  distance  between 
(6,  2)  and  (-3,  -4). 

3.  If  the  axes  are  rectangular,  find  the  distance  between  (a,  6)  and 
(c,  d). 


20 


ANALYTIC  GEOMETRY 


4.  Find  the  distance  between  the  points  whose  polar  coordinates  are 
(6,  20°)  and  (4,  2('-)),  where  2('-)  means  2  radians. 

5.  In  polar  coordinates  find  the  distance  between    (  —  3,    -  ]   and 


23.   Distance  between  two  points.     General  formula  in  rec- 
tangular coordinates. 

Let   Pi(aaj  2/i)  ^'^^   Pgfe  V-i)  be  two  points   in   rectangular 
coordinates,  and  let  d  =  PiPg-     Through 
^2  Pj  and  Pg  draw  lines  parallel,  respec- 
tively, to  the  X-  and  2/-axes  to  intersect 
in  M. 


Fig.  20. 


Then     d  =  Vp^ii^f '  +  j/pf. 
But  PiM=  X2  —  a^i,  JtfPa  =  2/2  —  2/i- 
.-.  d=  V(:«i- 0^27  + (2/1 -2/2)^ 


24.   Distance  between  two  points.     General  formula  in  polar 
coordinates. 


O2.  ^2) 


Fig.  21. 


Let  the  two  points  be  (rj,  ^1)  and  (rg,  dg)?  and  let  the  distance 
between  them  be  d.     (Fig.  21.) 


PROJECTIONS.     LENGTHS  AND  SLOPES  OF  LINES     21 

There  are  two  cases  to  consider :  according  as  the  difference 
between  the  vectorial  angles  is  less  than  or  greater  than  180°. 
In  the  first  case 

d2  =  rlfrl-2  r^r^  cos  {0^  -  O.), 
and  in  the  second  case 

d^  =  rf-^rl-2  r.r^  cos  [360°  -  (O^  -  ^i)]. 
These  reduce  to  the  one  form 

d  =  Vrf  +  r|  —  2  r^rg  cos  (^i  —  O2). 

EXERCISE  IX 

1.  Find  the  distance  between  (—4,  1)  and  (3,  5),  in  rectangular 
coordinates. 

2.  Find  the  distance  between  (3,  2)  and  (—  4,  —  5),  in  rectangular 
coordinates. 

3.  From  a  certain  point  0  three  other  points,  A,  J5,  and  O,  are  located 
as  follows  :  A  lies  3  mi.  N.  and  2^  mi.  E.  from  0,  B  lies  4  mi.  S.  and  1^ 
mi.  E.  from  0,  and  C  lies  5  mi.  W.  and  1^  mi.  N.  from  0.  Find  the  dis- 
tances between  the  points  A,  B,  and  C,  and  the  distance  of  each  of  the 
points  from  0  correct  to  hundredths  of  a  mile. 

4.  Find  the  distance  between  the  points  whose  polar  coordinates  are 
(4,  24°)  and  (-2,  40°). 

5.  Find  the  lengths  of  the  sides  of  the  triangle  whose  vertices  are 
(5,  —2),  (—4,  7)j  and  (7,  —3),  in  rectangular  coordinates. 

6.  Find  the  lengths  of  the  sides  of  the  triangle  whose  vertices  are 
(-2,  30°),  (4,  25°),  and  (5,  115°). 

25.   The  angle  which  one  line  makes  with  another. 

Definition.  The  angle  which  one  line,  L^,  makes  with 
another,  L2,  is  the 
angle,  not  greater  than 
180°,  measured  coun- 
ter-clockwise from  L2 
to  Li. 

Thus,  in  Eig.  22,  6 
is  the  angle  which  Li  Fia.  22. 


22  ANALYTIC  GEOMETRY 

makes  with  L^.     The  supplement  of  6  is  the  angle  which  L^ 
makes  with  L^. 

26.  Inclination  and  slope  of  a  line.  The  angle  which  a  line 
makes  with  the  x-axis,  or  with  any  line  parallel  to  the  a>axis, 
is  called  the  inclination  of  the  line. 

This  angle  is  to  be  measured  from  the  positive  direction  of 
the  avaxis  toward  the  positive  direction  of  the  ?/-axis. 

In  rectangular  coordinates,  the  slope,  or  gradient,  of  a  line  is 
the  ratio  of  the  change  of  the  ordinate  to  the  corresponding 
change  of  the  abscissa  of  a  point  moving  along  the  line.  It  is 
counted  positive  if  the  ordinate  increases  as  the  abscissa  in- 
creases ;  negative  if  the  ordinate  decreases  as  the  abscissa  in- 
creases. 

Thus,  if,  as  a  point  moves  along  a  line,  the  ordinate  increases 
one  unit  to  an  increase  of  3  units  in  the  abscissa,  the  line  has 
a  slope  of  i ;  while  if  the  ordinate  decreases  1  unit  to  an  in- 
crease of  3  units  in  the  abscissa,  the  line  has  a  slope  of  —  \. 

The  inclinations  of  these  lines  are,  respectively, 
6  =  tan-^i  =  18°  26', 

and  6'  =  tan-^  (-  i)  =  161°  34'.     (Fig.  23.) 


3 
Fig.  23. 

From  the  definitions  of  inclination  and  slope  it  follows  that 
slope  =  tangent  of  inclination, 

or,  designating  the  inclination  of  a  line  by  0  and  its  slope  by  m, 

m  =  tan  0. 

If  the  axes  are  not  rectangular,  the  equation, 
slope  =  tangent  of  inclination, 
is  taken  as  definition  of  the  slope. 


PROJECTIONS.    LENGTHS  AND  SLOPES  OF  LINES       23 


27.   Slope  of  a  line  through  two  points  in  terms  of  the  rec- 
tangular coordinates  of  the  points. 

Let  the  two  points  be  Pj  {x^,  y^)  and  P^  (x.,,  y^. 

Through  Pj  and  P^ 
draw  lines  parallel  to 
the  coordinate  axes  to 
meet  in  M.  (Fig.  24.) 
Then  whether  the  slope 
is  positive  or  negative 
its  value   is   given  by 


slope 


formula 
MP,     y. 


V\ 


PiM     HC2  —  oci 


If  Pi  is  the  higher  point,  then  slope  =^^^~'^-,  which  is  the 
same  as  the  above.  ^^  ~  ^- 

Therefore,  in  rectangular  coordinates,  the  slope  of  a  line  through 
two  points  is  the  difference  of  the  ordinates  of  the  points  divided 
by  the  corresponding  difference  of  the  abscissas  of  the  points. 

28.   Point  dividing  a  line  in  a  given  ratio.* 

Example.     To  find  the  point  which  divides  the  line  from 

(- 1,  5)  to  (6,  -  4)  in  the  ratio  3  :  2. 
Let  (-1,5)  be  P^,  (6,  -  4)  be  P„  and 

let  the  required  point  be  P{x,  y).    Then, 

by  hypothesis, 

PiP^3 

PP2     2* 

Through  P,  P^,    and  Po,   draw    lines 
parallel  to  the  axes  as  in  Fig.  25. 
Then,  from  similar  triangles, 

MP  ^P.P^S 

NP2     PP2     2' 

*  In  this  article  and  in  several  following  articles  the  word  "line"  is  fre- 
quently used  in  the  sense  of  "  segment  of  a  line,"  where  there  is  no  doubt  of 
the  meaning. 


Fig.  25. 


24 


ANALYTIC  GEOMETRY 


and 


I.e. 


and 


MP^_ 

_^i^_ 

3 

NP 

~PP2~ 

^2' 

x  +  l_ 

3 

6-x 

^2' 

5-y_ 

3 

jV  +  4 

~2' 

from  which  x  =  3^,  ?/  =  —  f . 

Hence  the  required  point  is  (3^,  —  |). 

29.  External  division.  The  point  Pis  said  to  divide  the  line 
P1P2  externally  when  it  lies  on  the  line  produced.     (Fig.  26.) 

The  segments  into  which  P  divides  P1P2  are  defined  to  be 
PiP  and  PP2.     The  first  segment  is  that  from  the  beginning  of 

the  line  to  the  point  of  division, 
and  the  second  segment  is  that 
from  the  point  of  division  to  the 
end  of  the  line.  Since  these 
segments  are  measured  in  oppo- 
site directions,  they  are  opposite 
in  sign.  Hence  their  ratio  is 
negative.  The  first  and  second 
segments  must  correspond  respectively  to  the  first  and  second 
terms  of  the  given  ratio  into  which  P  is  to  divide  PiP2' 

30.  Example  of  external  division. 

To  find  the  point  which  divides  the  line  from  (—1,  5)  to 
(4,  7)  in  the  ratio  —  |. 

Let  (—1,  5)  be  Pj,  (4,  7)  be  P2,  and  let  the  required  point 
be  P{x,  y).  ' 

Then  P^=JL 

PP.         3 

Since  P^P  must  be  numerically  less  than  PP^,  P  must  lie 
nearer  to  Pi  than  to  Pg,  i.e.  P  must  lie  on  the  portion  of  the 
line  extended  through  Pj. 


Fig.  26. 


PROJECTIONS.     LENGTHS  AND  SLOPES  OF  LINES     25 

Project  the  segments  so  as  to  obtain  their  a>  and  i/-projec- 
tions.     (Fig.  27.) 


Fig.  27. 


Then 

M^P     P,P 
PM^     PP^ 

2 
~3' 

and 

M,P,_P,P_ 
P^M^     PP^ 

2 
3* 

-•• 

x^l          2 
4-a;        '3' 

and 

5-.y          2 

from  which 

if=-ll, 

v  =  1 

Hence  the  required  point  is  (  —  11,  1). 


EXERCISE   X 


1.  Find  the  point  which  divides  the  line  from  (—3,  1)  to  (6,  —  5)  in 
the  ratio— |.  Ans.     (12,-9). 

2.  Show  that  the  point  which  bisects  the  line  joining  (xu  Vi)  and 

3.  Find  the  ratio  in  which  the  line  from  (2,  0)  to  (6,  0)  is  divided  by 
(1,  0)  ;  by  (5,  0)  ;  by  (9,  0). 

'A.    The  point  P(2,  k)  is  on  the  line  joining  Pi(—  2,  3)  and  Pa (4,  —  7)  ; 
find  the  ratio  into  which  P  divides  Pi  Pa,  and  the  value  of  k. 


26 


ANALYTIC  GEOMETRY 


31.  General  formulas  for  a  point  dividing  a  line  in  a  g'ven 
ratio. 

Let   the   line   from  Pi(it'i,  2/1)  to   Pal^^a?  2/2)  be   divided   by 
P(x,  y)  in  the  ratio  r :  1. 

There  are  three  cases  to  consider: 

(1)  P  between  Pj  and  P2, 

(2)  P  on  the  line  produced  through  P^ 

(3)  P  on  the  line  produced  through  Pg. 
In  (1)  r  may  have  any  positive  value, 

in  (2)  r  is  negative  and  numerically  less  than  1, 
in  (3)  r  is  negative  and  numerically  greater  than  1. 


M,     M 


Project  PiP  and  PP^  on  any  two  lines  parallel  to  the  axes. 
(Fig.  28.)     In  either  of  the  three  cases, 

¥iM-P^-r  and  ^^^-^^^-r 


or 


from  which 


-  =  r,  and  ^ — ^  =  r, 
^2  —  ^  2/2-2/ 


3C 


EXERCISE   XI 

1.  Find  the  point  which  divides  the  line  from  (-  1,  3)  to  (6,  —  £)  in 
the  ratio  3:2. 


PROJECTIONS.     LENGTHS  AND  SLOPES  OF  LINES     27 


2.  Find  the  point  which  divides  the  line  from  (3,  |)  to  (-  5,  8)  in  the 
ratio  —  f  ;  in  the  ratio  —  |. 

3.  Find  the  external  point  on  the  line  joining  Pi(a,  b)  and  P2(c,  d) 
which  is  n  times  as  far  from  Pi  as  from  F2. 

4.  Find  the  points  which  trisect  the  line  joining  Piixi,  yO  and 
P2(X2,  y-z)' 

5.  The  point  P  divides  the  line  PiPz  in  the  ratio  r  :  1 ;  trace  the  varia- 
tion in  r  as  P  moves  along  the  line  internally  from  Pi  to  P2,  then  on  from 
P2  to  00  ,  and  then,  changing  to  the  other  side  of  Pi,  comes  in  from  —  go 
to  Pi. 

32.   Angle  between  two  lines  of  given  slopes. 

Example.     Let  two  lines  L^  and  L.2  have  slopes 
respectively  ;  to  find  the  angle  which  Li 
makes  with  ig- 

Let  Li  and  L2  make  angles  Oi  and  O2 
respectively  with  the  a>axis,  and  let  the 
angle  which  L^  makes  with  io  he  <f>. 
Through  the  intersection  of  the  lines 
draw  a  line  parallel  to  the  a;-axis.  (Fig.  ■ 
29.)     Then  it  is  seen  that 

(ft  =  61  —  do- 

Hence  tan  <j>  =  tan  (^1  —  ^2) 

tan  ^,  —  tan  Oo 


Fig.  29. 


But 


1  -f-  tan  ^1  tan  O2 
tan  <9i  =  -  2,   tan  O2  =  3. 
-2-3 


.  tan  <^ 


1-2.3 


=  1. 


33.   The  angle  between  two  lines.     General  formula. 

Let  two  lines,  Li  and  L2,  have  slopes  mi  and  mg  respectively ; 
to  find  the  angle  which  Li  makes  with  L^. 

Let  the  angles  which  Lj  and  Lo  make  with  the  avaxis  be  Oi 
and  O2  respectively.     Then  m^  =  tan  O.1,  m^  =  tan  O.2. 


28 


ANALYTIC  GEOMETRY 


Let  <^  be  the  angle  which  Li  makes  with  L^. 
Through  the  intersection  of  L^  and  L.2  draw  a  line  parallel 
to  the  a>axis.     Then  (Fig.  30), 


case  (i), 
case  (ii), 

and  in  either  case 


Fig.  30. 

6i  >   ^2J 

<f)  =  0]  —  62', 

^1  <  ^2> 

tan  (j>  =  tan  (^1  —  O2) 

__   tan  $1  —  tan  $2 
1  +  tan  $1  tan  62 

_  mi  —  m2 
1  +  miTJia 

34.  Condition  for  parallel  lines,  and  for  perpendicular 
lines.  If  the  two  lines  of  the  preceding  article  are  paral- 
lel, tan  61  =  tan  O2,  and  hence  mj  =  mg.  If  the  two  lines  are 
perpendicular,  tan  <^  =  tan  90°  =  00 ,  and  hence  1  +  mimg  =  0. 

Conversely,  if  mi  =  ma,  tan  </>  =  0,  .  • .  </>  =  0,  and  therefore  the 
lines  are  parallel. 

If  l4-mim2  =  0,  tan<^  =  oo,  .'.  <^  =  90°,  and  therefore  the 
lines  are  perpendicular. 

Therefore,  the  condition  that  two  lines  of  slopes  mi  and  mg 
be  parallel   is   mi  =  m2 ;   the  condition  that  they  be  perpen- 


dicular is  1  +  Wjma 


0,  or  mi  = < 


PROJECTIONS.     LENGTHS  AND  SLOPES  OF  LINES     29 

EXERCISE  XII 

For  rectangular  axes.     Draw  a  figure  in  each  case. 

1.  Show  that  the  line  joining  (3,  2)  and  (—  2,  —  13)  is  perpendicular 
to  the  line  joining  (1,  3)  and  (4,  2). 

2.  Show  that  (—  1,  —  2),  (3,  2),  and  (—3,  0)  are  the  vertices  of  a 
right  triangle.     Find  the  other  angles. 

3.  Where  does  a  line  cut  the  x-axis  if  it  passes  through  (2,  —  3)  and 
is  parallel  to  the  line  through  (—1,  5)  and  (4,  —  2)  ? 

4.  A  line  is  drawn  perpendicular  to  the  line  through  Pi  ( —  2,  5)  and 
P2(4,  —  3)  at  its  middle  point ;  find  a  point  P  on  this  perpendicular 
whose  abscissa  is  3,  and  show  that  P  is  equidistant  from  Pi  and  P^. 

5.  The  vertices  of  a  triangle  are  (7,  4),  (—  2,  —  5),  and  (3,  —  10)  ; 
show  that  the  line  joining  the  middle  points  of  two  sides  is  parallel  to  the 
third  side,  and  is  half  as  long,  by  using  formulas  for  slope  and  distance. 

6.  Find  a  fourth  point  which  with  the  three  given  in  example  5  form 
the  vertices  of  a  parallelogram. 

7.  Two  lines,  Li  and  X2,  make  tan-i  2  and  tan-i  —  4  respectively 
wdth  the  ic-axis ;  find  the  angle  which  Ly  makes  with  L^. 

8.  The  vertices  of  a  triangle  are  Pi(— 1,  5),  P2(3,  —4),  and 
P3(6,  2)  ;  find  the  slopes  of  the  sides  and  the  angle  at  Pi. 

9.  Show  by  their  slopes  that  the  line  joining  (—3,  4)  and  (6,  1)  is 
parallel  to  the  line  joining  (7,  2)  and  (5,  |). 

10.  A  line  L  makes  an  angle  of  45°  with  the  line  through  (1,  1)  and 
(6,  8)  ;  find  the  slope  of  L  and  the  angle  which  it  makes  with  the  x-axis. 

11.  Li  passes  through  (4,  5)  and  (6,  —  3).     L-2.  is  perpendicular  to  Li ; 
find  the  slopes  of  Li  and  L^. 

12.  Zi  has  a  slope  m.     The  angle  which  L^  makes  v\dth  the  a>axis  is 
double  the  angle  which  L^  makes  with  the  a;-axis  ;  what  is  the  slope  of  L^  ? 

13.  The  slope  of  one  line  is  3.728  and  of  another  —  .324 ;   find  the 
acute  angle  between  them. 

14.  Find  the  slope  of  a  line  which  makes  an  angle  of  —  42°  with  a  line 
of  slope  .4364. 

15.  A  line  passes  through  (6,  —.  3)  and  has  a  slope  .324  ;  find  a  point 
on  the  line  with  abscissa  1.2. 

16.  A  line  cuts  the  x-axis  at  (a,  0)  and  makes  tan-i  m  with  the  ic-axis ; 
find  where  it  cuts  the  ?/-axis. 

17.  A  line  passes  through  (a,  0)  and  makes  tan-^w  with  a  line  of 
slope  n  ;  find  its  slope,  and  where  it  cuts  the  y-axis. 


30 


ANALYTIC  GEOMETRY 


III.     AREAS   OF   POLYGONS 

35.   Area  of  a  triangle  in  terms  of  the  coordinates  of  its 
vertices. 


Example  1.  To  find  the  area  of  a  triangle  whose  vertices 
in  rectangular  coordinates  are  Pi (—2,  3),  P2{4:,  —1),  and 
Ps(l,-6).      ■ 

Through  the  lowest  vertex,  Pg  (Fig.  31),  draw  a  line  parallel 
to  the  ic-axis,  and  from  the  other  vertices  drop  perpendiculars 

to  this  line,  meeting   it 
Y 


Pi  (-2,  3> 


-1) 


in  3/i  and  M^. 

Then    the     area     re- 
quired is  equal  to 

area  of  M^P^PiM^ 

—  area  of  P.M2P2 

—  area  of  MiP^Pi 
=  i  3f,M2(M,P,-^  M2P2) 


-\P,M2 

-iM,P, 

=  1.6(9  +  5) 
5 


M2P2 
M,P, 


-i-3 
-i-3 


=  21. 


Fig.  31. 


If     P1P2P3 

a   triangular 


9. 


represents 
field  to  a 


scale  of  1  space  =  n  ft., 
then  the  area  of  the  field  is  21  n^  sq.  ft. 

Example  2.  To  find  the  area  of  the  triangle  whose  vertices 
in  polar  coordinates  are  (3,  60°),  (-2,  125°),  and  (5,  215°). 

The  area  required  is  the  sum  of  the  areas  of  the  triangles 
OP2P,,  OPJ\,  0P,P2  (Fig.  32). 

The  area  of  a  triangle  is  equal  to  one  half  the  product  of  two 
sides  and  the  sine  of  the  included  angle. 


PROJECTIONS.     LENGTHS  AND  SLOPES  OF  LINES       31 

P,  (3,  60°) 


P,    (-2,  125°) 


Pg    (5,  215°) 

Fig.  32. 
.*.  the  required  area 

=  10P,'  OF, sin  P.OPs  +  iOPs'  OP2  sin  P3OP2 
+10P2-  OPisinP^OPi 
=  i  .  3  .  5  sin  155°  +  1 .  5  •  2  sin90°  +  i  •  2  .  3  sin  115° 
=  i  (15  sin  25°  4-10+6  cos  25°)  =  10.89. 

EXERCISE   XIII 

1.  Find  the  area  of  the  triangle  whose  vertices  in  rectangular  coordi- 
nates are  (3,  -  5),  (—  8,  6),  and  (9,  2). 

2.  Find  the  area  of  the  triangle  whose  vertices  in  polar  coordinates  are 

3.  Find  the  area  of  a  triangle  whose  vertices  in  rectangular  coordinates 
are  (0,  0),  (xi,  yi),  and  (x2,  y^). 

4.  Find  the  area  of  a  triangle  whose  vertices  in  polar  coordinates  are 
(0,  0),  (n,  ^i),  and  0-2,  ^2). 

5.  Find  the  area  of  the  quadrilateral  whose  vertices  in  rectangular 
coordinates  are  (-2,  5),  (7,  9),  (10,  -3),  and  (-6,  -9). 

36.  Area  of  a  triangle.  General  formula  in  rectangular 
coordinates.  Let  Pi(.Ti,  ?/i),  P-zipo,  1/2),  and  P,{x.^,  y,)  be  the  ver- 
tices of  a  triangle  in  rectangular  coordinates ;  to  find  the  area 
of  the  triangle. 


32 


ANALYTIC  GEOMETRY 


Through  the  lowest  vertex  (Pg  in  I'ig-  33)  draw  a  line  paral- 
lel to  the  a^-axis,  and  from  the  other  vertices  drop  perpendicu- 
lars to  this  line,  meeting  it  in  M^  and  M^.     Then 


M,P, 


area  of  triangle  P1P2P3 

=  area  of  trapezoid  M^P-^P^M^ 

+  area  of  triangle  P^P^M^ 

—  area  of  triangle  P2M^P^ 

=  ^{M,P,  +  JW3P3)  •  M,M,  +  i  P^M, .  M,P,  -  1  P^M, 

=  K  (2/1  -  :^2  +  2/3  -  2/2)  (^-3  -  a^i)  +  (a^i  -  ^2)  (2/1  -  2/2) 

-(^•3-a?2)(2/3-2/2)], 

or,  area  P^P^P^  =  J  (a'12/2  +  a^gl/s  +  ^32/1  —  ^iVs  —  ^iVi  —  ^3^2)- 
This  may  be  written  in  the  determinant  form 

i»i     2/1     1 

1  a;2    2/2     1 

^s    2/3     1 

In  Fig.  33  the  succession  of  subscripts  1,  2,  3,  is  obtained  by 
going  around  the  triangle  counter-clockwise.  If  the  points  had 
been  so  lettered  that  in  following  the  above  order  it  would  be 
necessary  to  go  around  the  triangle  clockwise,  the  area  would 
have  been  found  to  be  minus  the  above  expression. 

This  can  be  seen  to  be  true  by  exchanging  two  of  the  sub- 
scripts, say  1  and  2,  in  Fig.  33,  and  making  the  same  exchange 
in  the  formula.     The  change  in  the  figure  changes  the  order 


PROJECTIONS.     LENGTHS  AND  SLOPES  OF  LINES     33 

from  counter-clockwise  to  clockwise,  and  the  change  in  the 
formula  just  changes  the  sign  of  the  whole  expression. 

37.  Area  of  a  triangle.  General  formula  in  polar  coordi- 
nates. Let  Pi(ri,  $i),  P^ir^,  O^),  and  Pg^rg,  ^3)  be  the  vertices  of 
a  triangle  in  polar  coordinates  ;  to  find  the  area  of  the  triangle. 

Two  cases  are  to  be  distinguished,  according  as  the  pole  lies 
without  or  within  the  triangle.  The  second  case  will  occur 
only  when  the  difference  between  the  vectorial  angles  of  two 
of  the  vertices  is  greater  than  180°. 


Fig.  34. 


In  case  (1)  the  area  of  the  triangle  P^P^P^  is  equal  to  the  area 
of  triangle  OP^P-i  -\-  area  of  triangle  OPzPs  —  area  of  triangle 
OP1P3 

=  i  r^u  sin  (02  -  ^1)  +  i  r^r^  sin  (^3  -O^)-^  r^r^  sin  (^3  -  Oi) 
=  i  [ri^o  sin  (^2  -  ^1)  rh  r^r^  sin  (^3  —^^2)  4  ^31-1  sin  (^1  -  ^3)]. 

In  case  (2)  the  area  of  triangle  OP1P3  must  be  added  to  the 
areas  of  the  other  two  triangles,  instead  of  subtracted  from 
them,  as  in  case  (1) ;  but  area  of  OP1P3  is  here  equal  to 
i  ViV^  sin  [360°  —  (^3  —  ^1)]  which  is  equal  to  —  ^  ^Vg  sin  (^3  —  ^1). 
The  formula  for  the  area  of  the  triangle  sought  reduces  there- 
fore to  the  same  as  in  case  (1). 

Just  as  in  the  case  of  the  area  in  rectangular  coordinates,  the 
above  formula  would  give  the  negative  of  the  area  if  the  sub- 


34 


ANALYTIC   GEOMETRY 


scripts  were  so  arranged  that  in  following  the  order  1,  2,  S,  it 
would  be  necessary  to  go  around  the  triangle  clockwise. 

38.   Area   of   a  polygon.      General  formula  in  rectangular 
coordinates.     If  the  origin  be  one  of  the  vertices  of  a  triangle 
whose  other  vertices  are  Pi  (x^,  y^  and 
P2(x*2,  2/2),  the.  formula  for  the  area  of 
the  triangle  given  in  Art.  36  becomes 

provided  that  in  going  around  the  tri- 
angle counter-clockwise  the  vertices  are 
passed  in  the  order  P^,  P^,  and  0. 
This  area  of  the  triangle  OP1P2  may  be 
thought  of  as  generated  by  a  line  OP, 

initially  in  the  position  OPi,  turning  counter-clockwise  about 

0  to  the  final  position  OP2,  the  point  P  moving  along  the 

line  PiP2-     With  this  conception  of  the 

area,  it  must  be  noted  that  it  is  the  ab- 
scissa, x^,  of  the  initial  position,  Pj,  of 

P  which  comes  first  in  the  formula  for 

the  area,  ^(xiy2  —  .T22/1). 

If  the  line   OP  must  turn  clockwise 

from  the  position   OPi  to  the   position 

OP2,  then  the  expression   -J  (a^iif/2  —  ^^yi) 

is  equal  to  the  negative  of  the  area  of 

the  triangle  OP1P2. 

Let  ^(«i2/2  — ^22/1)  b®  den'oted  by  A.     Thus 

Consider  now  any  polygon  whose  vertices  in  rectangular  co- 
ordinates are  Pi(xi,  y^),  P2(x2,  y^,  •••  Pn^^m  2/n)j  the  vertices 
being  so  lettered  that  in  going  around  the  polygon  counter- 
clockwise the  vertices  are  passed  in  the  order  Pj,  Po,  •••  P„. 

For  definiteness  let  r?  =  6,  and  let  the  polygon  be  as  shown  in 
Fig.  37,  the  origin  being  outside  of  the  polygon.    Let  a  point  P 


PROJECTIONS.     LENGTHS  AND  SLOPES  OF  LINES      35 


start  at  Pj,  traverse  the  perimeter  of  the  polygon  counter- 
clockwise, and  return  to  P^.  The  line  OP  generates  in  order 
the  triangles  OP^Pi,  OP^P^,,  •••  OP^P^.  Now  the  area  gener- 
ated by  OP  which  lies  without 
the  polygon  is  generated  twice, 
with  OP  turning  once  clockwise, 
once  counter-clockwise;  or  else 
is  generated  four  times  with  OP 
turning  twice  clockwise,  twice 
counter-clockwise  ;  but  the  area 
within  the  polygon  is  generated 
once,  with  OP  turning  counter- 
clockwise; or  else  is  generated 
three  times,  with  OP  turning 
once  clockwise,  twice  counter- 
clockwise. Therefore  if  the  expression  A  be  formed  for  each 
of  the  triangles  OP1P2,  OP.Ps,  •••  OP^Pi,  and  their  sum  taken, 
all  the  area  generated  by  OP  will  be  cancelled  out  except  that 
within  the  polygon  and  that  area  will  be  counted  just  once. 
Therefore  the  area  of  the  polygon  is  equal  to 

i  (^m  -  ^'22/1  +  a?22/3  -  X^V2  +  ^sVi  -  ^42/3  +  ^42/5  "  ^oVa  +  ^52/6  "  ^sVs 

4-a:a2/i-^i2/6)- 


Fig.  37, 


0 


A  convenient  method  of  arranging  the  coordinates  for  the 
computation  of  the  area  is  as  follows :  Write  down  in  succession 
the  abscissas  of  the  vertices  taken  in  order  counter-clockwise  around 
the  polygon,  rejieating  the  first  abscissa  at  the  last;  under  the  ab- 
scissas write  the  corresponding  ordinates : 


Xi 

X2 

Xs 

x^ 

x^ 

Xq 

a?! 

2/1 

2/2 

2/3 

2/4 

2/5 

2/6 

2/1 

Then  multiply  each  abscissa  by  the  following  ordinate  and  take 
the  sum  of  the  terms  obtained;  multiply  each  ordinate  by  ihe  fol- 
lowing abscissa  and  take  the  sum  of  the  terms  obtained.  The  area 
is  half  of  the  first  sum  minus  half  of  the  second. 


36  ANALYTIC  GEOMETRY 


EXERCISE   XIV 

1.  The  vertices  of  a  polygon  taken  in  order  are  (6,  1),  (9,  —4), 
(3,  -  10),  (-  3,  -  5),  (-  6,  -  8),  (-  12,  0)  and  (-  4,  6)  ;  find  the  area 
of  the  polygon. 

2.  The  distances  north  of  a  fixed  east  and  west  line  of  four  points  A^ 
B,  C,  D  are  respectively  32.6  ft.,  65.1  ft.,  80.3  ft.,  51.7  ft.,  and  their  dis- 
tances east  of  a  fixed  north  and  south  line  are  respectively  25.3  ft.,  48.2 
ft.,  94.5  ft.,  106  ft.;  find  the  area  of  the  quadrilateral  ABCD. 

*  3.  The  distances  of  four  points  A,  B,  C,  D  from  a  point  O  are  respec- 
tively 120  ft.,  216  ft.,  320  ft.,  and  65  ft.,  and  their  directions  from  0  are 
respectively  E.  25°  N.,  N.  32°  W.,  S.  74°  W.,  E.  67°  S.  ;  find  the  area  of 
ABCD. 

4.  The  vertices  of  a  triangle  are  (3,  —  2),  (—4,  1),  and  (—8,  ~  5)  ; 
find  (a)  the  area,  (h)  the  lengths  of  the  sides,  (c)  the  slopes  of  the  sides, 
(d)  the  angles. 

5.  Show  (a)  by  the  lengths  of  the  sides,  (6)  by  the  slopes  of  the  sides, 
that  the  quadrilateral  whose  vertices  are  (1,  2),  (3,  —  2),  (—  1,  —  3),  and 
(—3,  1)  is  a  parallelogram.    Find  its  area. 

6.  Show  by  means  of  the  slopes  of  the  lines  that  the  line  joining  the 
middle  points  of  two  sides  of  any  triangle  is  parallel  to  the  third  side. 
Show  also  that  its  length  is  half  that  of  the  third  side. 

7.  The  vertices  of  a  triangle  are  Pi,  P2,  Ps  ;  find  the  point  which  di- 
vides the  line  from  Pi  to  the  middle  point  of  P2P3  in  the  ratio  2  : 1.  Show 
that,  using  either  of  the  vertices  in  like  manner,  the  same  point  is  obtained, 
and  hence  that  the  three  medians  of  a  triangle  meet  in  a  point. 

8.  In  the  formula  for  the  area  of  a  triangle  in  rectangular  coordinates, 
substitute  the  values  of  the  rectangular  coordinates  in  terms  of  the  polar 
coordinates  and  obtain  the  formula  for  the  area  of  the  triangle  in  terras  of 
polar  coordinates. 

9.  The  line  joining  (a,  &)  and  (c,  d)  is  divided  into  four  equal  parts  ; 
find  the  points  of  division. 

10.  Show  analytically  that  the  middle  points  of  the  sides  of  any  quad- 
rilateral are  the  vertices  of  a  parallelogram. 

11.  Prove  that  the  middle  point  of  the  line  joining  the  middle  points  of 
two  opposite  sides  of  any  quadrilateral  has  an  abscissa  equal  to  one  fourth 
the  sum  of  the  abscissas  of  the  vertices  of  the  quadrilateral,  and  find  the 
similar  relation  for  the  ordinates.     What  conclusion  can  you  draw  ? 


PROJECTIONS.     LENGTHS  AND  SLOPES  OF  LINES    37 

12.  The  point  (2,  k)  is  equidistant  from  (—  5,  7)  and  (3,  4)  ;  find  k. 

13.  The  point  (x,  y)  is  equidistant  from  (2,  —  1)  and  (7,  4)  ;    write 
the  equation  which  x  and  y  must  satisfy.     What  is  the  locus  of  (a;,  y)  ? 

14.  Express  by  an  equation  the  condition  that  the  point  (a;,  y)  is  dis- 
tant 5  from  (2,  3).     What  is  the  locus  of  the  point  (a:,  y)  ? 

15.  Show  that  the  line  joining  (4,  —  4,)  and  (—  2,  —  1)  is  perpendicu- 
lar to  the  line  joining  (3,  1)  and  (I,  —  3). 

16.  Find  the  angle  which  the  line  whose  slope  is  6,324  nlakes  with  the 
line  whose  slope  is  —  .657. 

17.  Find  the  slope  of  a  line  which  makes  an  angle  of  30°  with  a  line 
whose  slope  is  3. 

18.  The  line  Lx  makes  an  angle  of  40°  with  the  a;-axis,  and  the  line  L^ 
makes  an  angle  whose  tangent  is  2  with  Li ;  find  the  slope  of  L^. 

19.  If  Li  makes  tan-i  a  with  the  a;-axis,  and  L^  makes  tan-^  h  with  Lx^ 
find  the  slope  of  L». 

20.  The  angle  from  Lx  clockwise  to  L^  is  tan-i  (|),  and  the  angle  from 
L2  counter-clockwise  to  the  x-axis  is  tan-i(—  f)  ;  find  the  slope  of  Li. 


CHAPTER   III 

GRAPHICAL   REPRESENTATION   OF   A   FUNCTION; 
EQUATION    OF   A   LOCUS 

39.  Function  and  variable.  One  quantity  is  said  to  be  a 
function  of  a  second  quantity  when  to  every  value  of  the 
second  there  corresponds  one  or  more  values  of  the  first. 

Thus  in  the  equation  v  =  gt,  which  expresses  the  velocity  of 
a  body  falling  freely  in  a  vacuum  in  terms  of  the  time,  the 
velocity,  v,  is  a  function  of  the  time,  t. 

Again,  in  the  equation  pv  =  a  constant,  the  formula  which 
expresses  the  relation  between  the  pressure  and  volume  of  a 
gas  kept  at  constant  temperature,  either  of  the  quantities  p  or 
v  is  a  function  of  the  other  one. 

The  quantity  which  may  take,  or  to  which  may  be  assigned, 
arbitrary  values  is  called  the  independent  variable,  or  often 
simply  the  variable,  and  a  function  of  this  variable  is  often 
called  the  dependent  variable. 

According  to  the  above  definition  of  a  function  any  constant 
may  be  regarded  as  a  function  which  takes  the  same  value  for 
all  values  of  the  variable. 

Ii  to  every  value  of  the  variable  there  is  just  one  value  of 
the  function,  the  function  is  said  to  be  a  single-valued  function 
of  the  variable.  If  two,  three,  or  more  values  of  the  function 
exist  for  every  value  of  the  variable,  the  function  is  called  re- 
spectively a  double-valued,  triple-valued,  or,  in  general,  a 
multiple-valued  function  of  the  variable. 

Thus  in  v=32t,  v  is  a  single-valued  function  of  t,  and  in 
2/2  =  4  £c,  y  is  a  double- valued  function  of  x.  On  the  other 
hand,  a;  is  a  single-valued  function  of  y,  if  y  be  taken  as  the 
independent  variable. 

38 


GRAPHICAL  REPRESENTATION  OF  A  FUNCTION     39 

40.  The  graph  of  a  function.  It  is  not  always  possible  to 
express  by  means  of  an  equation  the  value  of  a  function  in 
terms  of  the  variable.  When,  however,  there  are  known 
several  pairs  of  corresponding  values  of  two  quantities,  one 
of  which  depends  upon  the"  other,  a  graphical  representation  of 
one  of  the  quantities  as  a  function  of  the  other  may  be  made 
which  will  exhibit  in  an  instructive  way  the  dependence  of  one 
of  the  quantities  upon  the  other. 

To  illustrate  this  consider  the  following  examples. 

Example  1.  It  was  found  that  when  a  certain  rod  of  steel 
was  subjected  to  tension,  the  values  of  the  extension  of  the  rod 
in  terms  of  the  tension  were  as  shown  in  the  following  table, 
in  which  T  is  the  number  of  pounds  of  tension  per  square 
inch  of  cross-section  of  the  rod  and  e  is  the  number  of  units  of 
extension  per  unit  length  of  the  rod,  the  initial  tension  being 
1000  lb. 

T      1000      5000  10,000  20,000  30,000  40,000  50,000  51,000 

c            0     .0003     .0009  .0019  .0030  .0040     .0053     .0056 

T  52,000  54,000  56,000  58,000  60,000  70,000  80,000 

£     .0058     .0064     .0075  .0089  .0113  .0272     .0500 

Take  the  values  of  €  as  abscissas  and  the  values  of  T  as 
ordinates  and  plot  the  points  representing  the  corresponding 
values  of  c  and  T.  Then  draw  a  smooth  curve  through  these 
points.  On  the  assumption  that  as  the  tension  changes  grad- 
ually, passing  through  all  values  between  the  first  and  last 
values  of  the  tension  that  are  given,  the  extension  also  changes 
gradually,  the  smooth  curve  through  the  plotted  points  may  be 
taken  as  a  graphical  representation  of  T  as  a  function  of  c  in 
the  sense  that  the  coordinates  of  any  point  on  the  curve  are 
corresponding  values  of  c  and  T. 

In  general  the  more  points  that  are  determined  by  known 
values  of  the  variables  the  more  accurately  will  the  curve 
represent  the  function.  Of  course,  too,  these  points  should  be 
somewhat  evenly  separated. 


40 


ANALYTIC  GEOMETRY 


Outside  the  range  of  values  given,  no  information  can  be 
drawn  from  the  curve  concerning  the  values  of  the  function 
for  a  given  value  of  the  variable. 


80,000 


60,000 


■-20,000 


jm 


.02  .03 

elojigation  injncheb 

Fig.  38. 


M 


.05 


The  curve  does  not  give  any  information  that  is  not  con- 
tained in  the  table,  but  gives  the  same  information  in  such  a 
way  as  to  bring  out  relations  that  are  not  readily  observed 
from  the  table. 

From  the  curve  it  is  seen  that  as  long  as  T  is  less  than 
about  50,000  the  extension  is  proportional  to  the  tension,  the 
points  of  the  curve  lying  on  a  straight  line  approximately,  but 
that  when  T  passes  through  the  value  50,000  the  extension 
increases  more  and  more  rapidly  as  T  increases. 

Also  the  value  of  T  corresponding  to  an  assumed  value  of  c 
may  be  found  approximately  from  the  curve  by  measuring  the 
value  of  the  ordinate  of  the  point  of  the  curve  which  has  the 
assumed  value  of  c  as  abscissa.  Likewise  the  value  of  e  corre- 
sponding to  an  assumed  value  of  T  may  be  found. 

Example  2.     The  following  table  shows  the  number  B  of 


GRAPHICAL  REPRESENTATION  OF  A  FUNCTION     41 

beats   per   minute  of  a  simple  pendulum  of  length  L  centi- 
meters for  certain  values  of  L : 

L      10      12      15      20      25     30   40    50    60    70    80   90    100 
B    190    172    154    136    120    110   95    85    78    72    67    63      60 


200 

\ 

\ 

\ 

\  150 

> 

\ 

\ 

a. 

\ 

2 

K 

^  100 

\ 

N 

\ 

I 

■^ 

^ 

>^ 

■-J 

s. 

'S 

p— 

- 

50 

25  50 

LENGTH  IN  CM. 

Fig.  39. 


75 


Take  the  values  of  L  as  abscissas  and  the  values  of  B  as 
ordinates  and  plot  the  points  representing  the  corresponding 
values  of  L  and  B.  The  curve  drawn  through  these  points 
shows  graphically  the  manner  in  which  B  depends  upon  L. 

It  also  enables  one  to  pick  out  approximately  the  value  of  B 
for  a  given  value  of  L  within  the  limits  given,  or  the  value  of 
L  for  a  given  value  of  B. 

41.  Equation  of  a  locus.  In  each  of  the  two  preceding  ex- 
amples a  curve  was  drawn  such  that  the  coordinates  of  all 
of  its  points  were  corresponding  values  of  the  function  and 
variable,  but  no  equation  was  found  which  expressed  the 
dependence  of  the  function  upon  the  variable. 

In  each  of  the  examples  to  be  next  studied  some  simple 
locus  of  points  will  be  considered,  and  the  equation  which 
expresses  the  dependence  of  the  ordinate  of  any  point  of  the 


42 


ANALYTIC  GEOMETRY 


locus  Tipon  the  abscissa  of  the  point  will  be  derived.  This 
equation  will  be  known  as  the  equation  of  the  locus. 

Definition.  The  equation  of  a  locus  is  an  equation  between 
the  coordinates  of  any  point  of  the  locus. 

The  locus,  on  the  other  hand,  is  called  the  locus  of  the  equa- 
tion. 

42.  Two  fundamental  problems.  The  two  fundamental  prob- 
lems of  Plane  Analytic  Geometry  are  : 

(1)  Having  given  a  locus  of  points  determined  by  certain 
geometric  conditions,  to  find  the  equation  of  that  locus. 

(2)  Having  given  an  equation  in  two  variables,  to  find  by  a 
study  of  the  equation  the  form  and  properties  of  the  locus 
which  it  represents. 

In  this  chapter  some  examples  illustrating  the  methods  of 
finding  the  equation  of  a  given  locus  will  be  considered,  and 
in  the  next,  chapter  some  methods  of  obtaining  the  locus  when 
the  equation  is  given  will  be  studied. 

43.  Illustrations.  Example  1.  Consider  the  locus  of  a  point 
which  moves  along  the  straight  line  passing  through  the  points 

Pi  (3,    -1)   and    P^ 
Y 


Pa  (-5.  4) 


(-5,  4).  If  any 
point  P(x,  y)  be 
taken  on  this  line, 
the  value  of  the  or- 
dinate clearly  de- 
pends upon  the  value 
of  the  abscissa  of 
the  point.  That  is, 
1?/  is  a  function  of  x. 

To  find  the  law, 
or  equation,  which 
expresses  the  depen- 
dence of  y  upon  X,  draw  through  P,  Pj,  and  P^  lines  parallel  to 
the  axes  to  form  the  triangles  PM^P^  and  PiM^P^  as  in  Fig.  40. 


P(x,  1/) 


GRAPHICAL  REPRESENTATION  OF  A  FUNCTION       43 


Then  by  similar  triangles 


M.Po 


I.e. 


M,P{ 

(_5)_4-(-l) 


y 


x-3  -1 

which  reduces  to  5  aj  +  8  ?/  =  7.  (1) 

If  P{x,  y)  is  a  point  not  on  the  line  through  Pj  and  P^,  the 
triangles  PM^P^  and  P^M^P^  are  not  similar,  and  equation  (1) 
does  not  hold.  Hence  equation  (1)  holds  for  all  points,  on  the 
line  and  for  no  others.     It  is  therefore  the  equation  of  the  line. 

The  equation  may  be  solved  for  y  and  written 

The  equation  is  the  law  of  the  dependence  of  y  upon  x.  It 
may  be  stated  as  follows :  The  ordinate  of  any  point  on  the 
straight  line  jmssing  through  (3,  —1)  and  (—5,  4)  is  equal  to 
—  I  of  the  abscissa  of  the  point  plus  |. 

Equation  (1)  might  also  be  solved  for  x,  which  would  ex- 
press X  as  a  function  of  y. 

Example  2.  Consider  the  locus  of  a  point  which  moves  so 
as  to  keep  always  at  a  distance  6  from  the  point  Pi (3,  2). 

The  locus  is  a  circle  with 
radius  6  and  ■  with  center  at 
(3,  2). 

Here  again  the  value  of  the 
ordinate  of  any  point  on  the 
locus  is  a  function  of  the  ab- 
scissa of  the  point.  To  find 
the  law  that  expresses  the 
ordinate  as  a  function  of  the 
abscissa,  consider  any  point 
P  (x,  y)  on  the  circle.  The  con- 
dition that  Pmust  fulfill  is  that  Fig,  41. 


PiP=6. 


44 


ANALYTIC  GEOMETRY 


Now  P^P=  -^{x  -  3f  +  {y-  2)2. 

0^^-3)2  4- (2/ -2)2  =  36.  (2) 

Since  eq.  (2)  is  true  for  all  points  on  the  circle  and  for  no 
others,  it  is  the  equation  of  the  locus. 

If  the  equation  be  solved  for  y,  the  result  is 


2/  =  2±  V36-(a;-3)2. 

This  equation  expresses  2/  as  a  function  of  x. 
Since  there  are  two  values  of  y  for  every  value  of  a;,  2/  is  a 
double-valued  function  of  x. 

Equation  (2)  might  be  solved  for  x,  and  x  be  thus  expressed 
as  a  function  of  y. 

Example  3.    A  point  moves  in  the  plane  so  as  to  keep  equi- 
distant from  Pi(3,  —  2)  and  Pii—  4,  7)  ;  to  find  the  equation  of 

the  locus. 

To  find  the  equation  of  the 
locus,  one  must  express  by 
means  of  an  equation  which 
contains  the  coordinates  of  any 
point  of  the  locus  that  geomet- 
ric condition  which  is  satisfied 
by  all  points  of  the  locus  and 
by  no  others.  This  property 
is  expressed  by  the  equation 
P^P=P^P. 

Expressed  in  terms  of  the   co- 
ordinates of   the  point  P,  this  equation  becomes 

V(a^-3)2  +  (2/  +  2)2  =  V(x'  +  4)2  +  (2/-7)l  (1) 

Squaring  both  members,  cancelling,  and  collecting,  there  results 
7a;-92/-+-26  =  0,  (2) 

which  is  the  desired  equation  of  the  locus.  For  all  values  of  x 
and  y  that  satisfy  (1)  also  satisfy  (2).  In  retracing  the  steps 
from  (2)  to  (1),  a  double  sign  is  introduced  which  would  give 


Fig.  42. 


GRAPHICAL  REPRESENTATION  OF  A  FUNCTION     45 


P^P  =  ±  P^P.  But  as  P^P  and  P^P  are  positive  distances,  the 
equation  containing  the  minus  sign  has  no  geometric  signifi- 
cance. Equations  (1)  and  (2)  therefore  are  satisfied  by  pre- 
cisely the  same  points. 

The  locus  is  known  from  plane  geometry  to  be  the  straight 
line  which  is  perpendicular  to  PiP^  at  its  middle  point. 

Example  4.  A  point  moves  so  that  the  sum  of  its  distances 
from  Pi (4,  0)  and  PgC—  4,  0)  is  always  equal  to  10  ;  to  find  the 
equation  of  the  locus. 

Let  P  (x,y)  be  any 
point  of  the  locus.  The 
geometric  condition  sat- 
isfied by  all  points  of 
the  locus  and  by  no 
others  is  expressed  by 
the  equation 

P2P+PiP=10. 

Expressed  in  terms  of 

the  coordinates  of  the  point  P,  this  becomes 

V(a;-4y-f-2/'  -f-  ^{x-\-4:)\+f  =  10. 

When  freed  from  radicals,  this  equation  becomes 

9  ar^-f  25  2/^  =  225. 

This  is  the*  equation  of  the  locus.  It  will  be  shown  in  Art.  83 
that  no  new  points  are  introduced  into  the  locus  by  squaring. 

A  point  which  moves  so  that  the  sum  of  its  distances  from 
two  fixed  points  is  constant,  describes  an  ellipse. 

The  above  locus  is  therefore  an  ellipse. 

Points  of  the  locus  may  be  obtained  by  describing  arcs  with 
Pi  and  P2  as  centers  and  radii  whose  sum  is  10.  The  inter- 
sections of  two  such  arcs  are  points  of  the  locus. 

Example  5.  A  point  moves  so  that  the  difference  of  its  dis- 
tances from  Pi(5,  0)  and  P2(— 5,  0)  is  8;  to  find  the  equation 
of  the  locus. 


Fig.  43. 


46 


ANALYTIC  GEOMETRY 


Fig.  44. 


Let  P(x,  y)  be  any  point  of  the  locus. 

The  geometric  condition  satisfied  by  all  points  of  the  locus 

and    by     no    other 
Y  V    points   is  then 

This  equation  when 
expressed  in  terms 
of  X  and  y  and  freed 
from  radicals  re- 
duces to 

9a72_16/  =  144, 

which   is   the  equa- 
tion   of    the    given 
locus.       It   will   be 
shown  in  Art.  87  that  no  new  points  are  introduced  into  the 
locus  by  squaring. 

A  point  which  moves  so  that  the  difference  of  its  distances 
from  two  fixed  points  is  constant,  de- 
scribes an  hyperbola. 

The  above  locus  is  therefore  an 
hyperbola. 

Points  of  the  locus  may  be  obtained 
by  describing  arcs  with  P^  and  P^  a-s 
centers  and  radii  whose  difference  is  8. 
The  points  of  intersection  of  two  such 
arcs  are  points  of  the  locus. 

Example  6.  A  point  moves  so  that 
it  remains  always  equidistant  from  Pj 
(6,  0)  and  the  2/-axis ;  to  find  the  equa- 
tion of  the  locus. 

Let  P{x,  y)  be  any  point  of  the  lo- 
cus. From  P  draw  PM  perpendicular 
to  OY.     Then  the  geometric  condition  to  be  satisfied  by  P  is 


[ 


GRAPHICAL  REPRESENTATION  OF  A  FUNCTION     47 
expressed  by  the  equation 

MP=PyP. 

Expressed  in  terms  of  the  coordinates  of  P{Xj  y)  this  is 


x  =  ^{x-Qf  +  f,  (1) 

which  on  squaring  reduces  to 

2/2  =  12aj-36.  (2) 

This  is  the  equation  of  the'  locus. 

That  no  new  points  were  introduced  into  the  locus  by  squar- 
ing eq.  (1)  may  be  seen  as  follows :  Any  values  of  x  and  y  that 
satisfy  (1)  also  satisfy  (2),  but  there  are  values  of  x  and  y  that 
satisfy  (2)  that  do  not  satisfy  (1).  For  in  retracing  the  steps 
from  (2)  to  (1)  a  double  sign  is  introduced ;  i.e.  given  eq.  (2), 
there  follows 

x^±^{x-QY  +  y\ 

Now  it  is  evident  geometrically  that  no  point  can  be  equi- 
distant from  the  ^/-axis  and  (6, 0)  and  have  its  abscissa  negative. 
Therefore  only  the  plus  sign  can  be  used.  Therefore  all  points 
whose  coordinates  satisfy  (2)  also  satisfy  (1).  No  real  values 
of  X  and  y  could  therefore  have  been  introduced  into  eq.  (1) 
by  squaring. 

A  point  which  moves  so  as  to  keep  equidistant  from  a  fixed 
point  and  a  fixed  straight  line  describes  a  parabola. 

The  above  locus  is  therefore  a  parabola. 

44.  Method  of  finding  the  equation  of  the  locus  of  points 
which  satisfy  a  given  condition.  In  finding  the  equation  of 
the  locus  of  points  satisfying  a  given  condition,  a  certain 
method  was  followed  in  the  preceding  examples.  This 
method  will  suffice  for  finding  the  equation  of  the  locus  of 
points  satisfying  any  condition,  if  that  condition  can  be  ex- 
pressed by  means  of  an  equation.  The  method  may  be  formu- 
lated as  follows : 


48  ANALYTIC  GEOMETRY 

To  find  the  equation  of  the  locus  of  points  which  satisfy  a 
given  condition, 

(1)  Assume  any  point  P  on  the  locus. 

(2)  Write  the  equation  tvhich  expresses  the  condition  that  P 
must  satisfy. 

(3)  Express  this  equation  in  terms  of  the  coordinates  of  P 
and  simplify  the  equation. 

45.  Intercepts  of  a  locus  on  the  axes.  The  abscissa  of  a 
point  where  a  locus  cuts  the  a>axis  is  called  an  jr-intercept  of 
the  locus.  The  ordinate  of  a  point  where  a  locus  cuts  the 
2/-axis  is  called  a /-intercept  of  the  locus. 

If  the  equation  of  the  locus  is  known,  the  ic-intercepts  may 
be  found  by  letting  y  equal  zero  in  the  equation  and  solving 
the  resulting  equation  for  x.  Likewise  the  ^/-intercepts  may 
be  found  by  letting  x  equal  zero  in  the  equation  and  solving 
the  resulting  equation  for  y. 

EXERCISE  XV 

Derive  the  equations  of  the  following  loci.  Find  the  intercepts  of  the 
loci  on  the  axes.     Plot  the  loci. 

1.  A  straight  line  through  (1,  4)  and  (—  6,  7). 

2.  A  straight  line  through  the  origin  making  an  angle  of  60*^  with  the 
aj-axis. 

3.  The  X-axis.     The  y-axis.     A  parallel  to  the  a;-axis  through  (5,  2). 

4.  A  straight  line  through  (.3,  —  5)  with  slope  2. 

5.  A  straight  line  through  (a,  0)  and  (0,  6). 

6.  A  straight  line  through  (0,  h)  with  slope  m. 

7.  A  circle  with  radius  5  and  center  at  (2,  —  4). 

8.  A  circle  with  center  at  (-  6,  4)  and  passing  through  (3,  1). 

9.  A  circle  with  the  ends  of  a  diameter  at  (5,  -  6)  and  (3,  12). 

10.  A  circle  with  center  at  (h,  k)  and  radius  r. 

11.  A  circle  with  center  at  the  origin  and  radius  r. 

12.  A  circle  tangent  to  both  axes  and  radius  r. 

13.  A  circle  tangent  to  the  ^/-axis  at  the  origin  and  radius  r. 


GRAPHICAL  REPRESENTATION  OF  A  FUNCTION     49 

14.  The  locus  of  a  point  which  moves  so  that  the  sum  of  its  distances 
from  (0,  3)  and  (0,  -  3)  is  8. 

15.  The  locus  of  a  point  which  moves  so  that  the  difference  of  its 
distances  from  (0,  3)  and  (0,  —  3)  is  4. 

16.  The  locus  of  a  point  which  moves  so  as  to  remain  always  equi- 
distant from  the  point  (0,  —  4)  and  the  x-axis. 

17.  The  locus  of  a  point  which  moves  so  that  the  sum  of  its  distances 
from  (3,  2)  and  (-  6,  1)  is  12. 

18.  The  locus  of  a  point  which  moves  so  that  the  difference  of  its  dis- 
tances from  (2,  3)  and  (—5,  —  1)  is  6. 

19.  The  locus  of  a  point  which  moves  so  as  to  keep  equally  distant 
from  (—3,  4),  and  the  line  parallel  to  the  y-axis  through  (8,  6). 

20.  The  perpendicular  bisector  of  the  line  joining  (1,  7)  and  (8,  2). 

21.  A  column  of  concrete  50  in.  long  was  compressed  longitudinally 
and  the  following  numbers  obtained,  in  which  P  ■=  number  of  pounds 
compression  per  square  inch  of  cross  section  of  the  column,  and  e  =  num- 
ber of  inches  of  compression,  the  initial  load  being  100  lb.  per  square 
inch. 

P  100  150  200  300  400  600        550 

e  0  .0007  .0015  .0034  .0057  .0080  .0093 

P  600  600  650  700  800  900      1000           ,          -  .,    , 

e  .0108  .0112  .0121  .0139  .0175  .0221  .0275        ^°  "°^^   *^  ® 

Make  a  graph  which  shows  P  as  a  function  of  c,  and  get  what  informa- 
tion you  can  from  the  curve. 

22.  A  steel  rod  of  diameter  .564  in.,  length  3  in.,  was  subjected  to  a 
tensile  force.     The  following  measurements  were  made,  in  which 

P  =  number  of  pounds  tension  per  square  inch  of  cross  section  of  the  rod, 
X  =  number  of  inches  extension,  the  initial  load  being  1000  lb.  per  square 
inch. 

P  1000        5000      10,000      20,000    30,000    40,000    36,000    37,000 

X  0        .0003        .0008        .0018       .0028       .0039       .0058      .0072 

P  38,000  39,000  40,000  41,000  42,000  44,000  46,000  50,000 
X  .0114        .0559        .0596        .0615       .0669       .0800       .0905       .1210 

Make  a  graph  which  shows  P  as  a  function  of  X.  What  information  do 
you  get  from  the  curve  ? 

E 


50  ANALYTIC  GEOMETRY 

23.  The  following  measurements  were  taken   in  an  experiment  in 
which  an  india  rubber  cord  was  stretched  by  hanging  a  weight  to  its  end. 

W  =  weight  in  kilograms,     L  =  length  in  centimeters. 


w 

0 

.5 

1.0 

1.5 

2.0 

2.5 

3.0 

3.5 

L 

10 

10.1 

10.3 

10.6 

10.9 

11.3 

11.7 

12.2 

W 

4.0 

4.5 

5.0 

5.5 

6.0 

6.5 

7.0 

7.5 

L 

12.7 

ia.3 

13.9 

14.6 

15.3 

16.1 

16.9 

17.9 

Make  a  graph  which  shows  TF  as  a  function  of  L. 

24.  In  Ex.  23  reduce  W  to  pounds  and  L  to  inches,  and  draw  the 
graph.  How  does  the  curve  compare  with  that  of  Ex.  23?  By  what 
choice  of  scale  units  could  you  make  the  two  curves  coincide  ? 


CHAPTER   IV 


LOCUS  OF  AN  EQUATION 

46.  The  second  fundamental  problem.  In  the  preceding 
chapter  some  equations  of  simple  loci  were  obtained  from  the 
geometric  conditions  which  the  points  of  the  loci  satisfied.  In 
this  chapter  the  converse  problem  of  finding  the  locus  when 
the  equation  is  given  will  be  considered  for  some  simple  equa- 
tions. 

Illustrations.  Example  1.  To  find  the  locus  of  the 
equation  y  =  2x-\-l. 

Any  number  of  points  whose  coordinates  satisfy  this  equa- 
tion may  be  found;  for  any  value  may  be  assigned  to  x  and  a 
corresponding  value  for  y  computed  from  the  equation.     A  few 
corresponding  values  so  obtained  are  : 
X    0,  1,  2, 
1,  3,  5, 


4,  -3,   --!/, 


y    1,  6,  5,  9,  -5,  -12. 

Plot  the  points  determined  by  these  pairs  of  values  of  x  and  y. 
They  seem  to  lie  on  a  straight  line. 

That  the  locus  of  the  equation  is  a  straight  line  may  be 
proved  as  follows : 

Draw  a  straight  line  through  two  points 
whose  coordinates  satisfy  the  equation,  as  Pj 
(0,  1)  and  P,(2,  5).  (Fig.  46.)  Take  any 
point  F(x,  y)  on  this  line  and  through  it 
draw  a  line  parallel  to  the  a^axis.  Erom  Pi 
and  P.J  drop  perpendiculars  to  this  line, 
meeting  it  in  Jfj  and  M^. 

Then  from  similar  triangles,  PM^P^  and 
PM,P,, 

M^P      M,P'        x-2     x-0' 
51 


V 

'  /p 

/' 

T 

__i 

1 

X 

r/ 

^4f 

ii._|m^- 

1 

Fig.  46. 


52 


ANALYTIC  GEOMETRY 


which  reduces  to 

y  =  2x  +  l. 

This  equation,  therefore,  holds  for  every  point  on  the  line. 

Conversely,  all  points  whose  coordinates  satisfy  the  equation 
lie  on  the  line ;  for  if  a  point  P(x,  y)  be  taken  not  on  the  line, 
the  triangles  PM^P^  and  PM^P^  are  not  similar,  and  hence  the 
above  equation  does  not  hold. 

Hence  the  equation  y  —  2  x-\-l  is  satisfied  by  all  points  on 
the  straight  line  through  (0,  1)  and  (2,  5)  and  by  no  others. 
The  line  is  therefore  the  locus  of  the  equation. 

Example  2.     To  find  the  locus  of  the  equation, 

This  equation  may  be  brought  into  a  form  like  that  of  eq.  (2) 
of  Art.  43,  by  completing  the  squares  in  the  terms  containing  x 
and  in  those  containing  y  as  follows, 

a:^  _  6  aj  4".  9  +  ?/2  +  B  y  +  1 6  =  24  +  9 -f  1 6, 
or  (a;-3)2  +  (y  +  4)2  =  49. 

Now  the  left-hand   member  of  this  equation  is  equal  to  the 

square  of  the  distance  from 
(x,  y)  to  (3,  —4),  and  the 
equation  therefore  states 
that  this  distance  is  equal 
to  7.  Hence  (x,  y)  must 
lie  on  the  circumference 
of  a  circle  with  center  at 
(3,—  4)  and  radius  7. 

Moreover,  the  coordinates 

of  any  point  on  this  circle 

satisfy  the  equation.    Hence 

the  circle  is  the  locus  of  the 

Fig.  47.  eqniation.     (Fig.  47.) 

Example  3.     To  plot  the  locus  of 


r 


4  X. 


LOCUS  OF  AN  EQUATION 


53 


The  following  pairs  of  values  of  x  and  y  are  obtained  by  ar- 
bitrarily assigning  values  to  x  and  computing  the  corresponding 
values  of  y. 
X    0,       1,       1,  2,  3,      4,  5,  6,  10, 

y     0,   ±1,   ±2,   ±V8,   ±Vi2,   ±4,   ±V20,   ±V24,   ±V40: 

From  the  equation  the  following  facts  are  readily  seen  to  be 
true: 

(1)  If  X  is  negative,  y  is  imaginary ;  therefore  no  part  of  the 
locus  lies  to  the  left  of  the  y-Sixis. 

(2)  Every  positive  value  of  x  gives  two  values  of  y  which 
differ   only    in    sign ;    there- 


fore the  points  of  the  locus 
lie  in  pairs  such  that  the 
X-axis  bisects  at  right  angles 
the  lines  joining  the  pairs. 

(3)  As  X  increases,  the 
positive  value  of  y  also  in- 
creases, and  as  x  becomes 
infinite,  y  also  becomes  in- 
finite ;  the  locus  therefore 
recedes  indefinitely  from 
both  axes  as  x  increases  in- 


Y 

^^-^^ 

^^^ 

^^ 

^^     ' 

/ 

z 

X 

0  ^ 

s 

\     ^ 

^:^    : 

"^^^ 

'^^^ 

FiQ.  48. 


definitely. 

(4)  A  small  change  in  x  makes  a  small  change  in  y. 

The  part  of  the  locus  which  lies  in  the  first  quadrant  may, 
therefore,  be  thought  of  as  generated  by  a  moving  point  which, 
starting  at  the  origin,  moves  along  a  curve  gradually  rising  as 
the  point  moves  to  the  right  and  passing  through  the  above 
calculated  points. 

The  part  of  the  locus  which  lies  below  the  a^axis  could  be 
obtained  from  that  above  the  a;-axis  by  folding  the  upper  part 
of  the  plane  over  upon  the  lower  part,  using  the  a;-axis  as  an 
axis  of  revolution. 

The  locus  is  therefore  approximately  the  curve  of  Fig.  48. 


54  ANALYTIC  GEOMETRY 

EXERCISE   XVI 

Prove  that  the  locus  of  each  of  the  equations  from  1  to  6  is  a  straight 
line.     Find  the  intercepts  of  the  lines  on  the  axes  and  draw  the  lines. 

1.  3x-4y^6.  ^     ^  +  y-l  4.   y  =  7a;  +  3. 

2.  2x  +  5y  =  12.  '34~'  5.   ix-Sy  +  9  =  0. 
Prove  that  the  locus  of  each  of  the  following  equations  is  a  circle,  and 

find  the  center. and  radius. 

6.  x^-{-y^-4x  =  0.  9.   x^-\-y^-2ax-2by  =  r^-a^-b^. 

7.  x^-\-y2-8x  \-2y  =  S.  10.    x:^  +  y^  +  x -Sy  =  1. 

8.  x^  +  y^  =  r^.  11.   x2  +  y^-2ax  =  0. 
Plot  the  loci  of  the  following  equations  : 

12.  2/2  =  4(x-2).  14.   a;2  =  8(y-4).       16.   x^=-y. 

13.  x^  =  6y.  15.    y^  =  -4x.  17.    a:-3  =  2  (y  +  1)2. 

18.  y^  =  mx,  letting  m  =  j\,  1,  4,  16,  100,  —  1,  -  100. 

19.  x^  =  my,  letting  w  take  different  values. 

20.  x2  +  4y2  =  i6.  21.    x^-4y^=l6. 

47.  Symmetry.  Before  taking  up  more  difficult  problems 
in  loci  it  will  be  well  to  discuss  briefly  the  subject  of  symme- 
try of  a  curve  with  respect  to  a  line  9,nd  with  respect  to  a  point. 

Two  points  are  said  to  be  symmetric  with  respect  to  a  given 
line  when  the  given  line  bisects  at  right  angles  the  line  joining 
the  two  points. 

Two  points  are  said  to  be  symmetric  with  respect  to  a  given 
point  when  the  given  point  bisects  the  line  joining  the  two 
points. 

A  locus  of  points  is  said  to  be  symmetric  with  respect  to  a 
given  line  when  all  points  of  the  locus  lie  in  pairs  which  are 
symmetric  with  respect  to  the  given  line. 

The  line  is  then  called  an  axis  of  symmetry. 

A  locus  of  points  is  said  to  be  symmetric  with  respect  to  a 
given  point  when  all  points  of  the  locus  lie  in  pairs  which  are 
symmetric  with  respect  to  the  given  point. 

The  given  point  is  then  called  a  center  of  sjrmmetry 

Illustrations,    (a)  The  points  (x,  y)  and  (—  x,  y)  are  sym- 


LOCUS  OF  AN  EQUATION  55 

metric  with  respect  to  the  y-a,xis,  the  points  (x,  y)  and  (x,  —  y) 
are  symmetric  with  respect  to  the  aj-axis,  and  the  points  {x,  y) 
and  (—  X,  —  y)  are  symmetric  with  respect  to  the  origin. 

(p)  In  2/^  =  4  X,  if  (Xj  y)  is  a  point  of  the  locus,  so  also  is 
{x,  —y)\  for  if  the  coordinates  of  either  point  satisfy  the 
equation,  so  do  the  coordinates  of  the  other.  The  locus  is 
therefore  symmetric  with  respect  to  the  ic-axis. 

(c)  In  x^-\-4:y'^  =  16,  if  {x,  y)  is  a  point  on  the  locus,  so  are 
{—X,  y),  (x,  —y),  and  (—a*,  —y)',  for  if  the  coordinates  of 
the  first  point  satisfy  the  equation,  so  do  the  coordinates  of 
each  of  the  other  points.  The  locus  is  therefore  symmetric 
with  respect  to  the  y-axis,  with  respect  to  the  avaxis,  and  with 
respect  to  the  origin. 

48.  Tests  for  symmetry  with  respect  to  the  coordinate  axes 
and  the  origin.  If  an  equation  is  such  that  it  is  unchanged 
by  replacing  a;  by  —  x,  the  locus  of  the  equation  is  symmetric 
with  respect  to  the  ^/-axis.  For,  whatever  value,  say  x^,  be 
given  to  x,  the  resulting  equation  which  determines  the  cor- 
responding value,  or  values,  of  y  will  be  the  same  equation  as 
that  obtained  by  substituting  —  x^  for  x.  Hence  Xi  and  —  ajj 
give  the  same  values  of  y. 

Similarly,  if  replacing  y  hy  —  y  leaves  the  equation  un- 
changed, the  locus  is  symmetric  with  respect  to  the  avaxis. 

If  replacing  a;  by  —  a;  and  y  hj  —y  leaves  the  equation  un- 
changed, the  locus  is  symmetric  with  respect  to  the  origin. 

In  particular,  if  an  equation  contains  only  even  powers  of  x, 
the  locus  is  symmetric  with  respect  to  the  y-axis.  If  it  con- 
tains only  even  powers  of  y,  the  locus  is  symmetric  with  respect 
to  the  avaxis.  If  the  terms  of  an  equation  are  all  of  even 
degree,  or  are  all  of  odd  degree  in  x  and  y,  the  locus  is  sym- 
metric with  respect  to  the  origin.  (In  applying  this  last  test 
a  constant  term  must  be  considered  as  of  even  degree.) 

49.  Discussion  of  an  equation.  When  it  is  desired  to  plot 
the  locus  of  an  equation  in  two  variables,  it  is  well  to  discover 


56  ANALYTIC  GEOMETRY 

as  many  properties  and  facts  concerning  the  locus  as  one  can 
by  a  study  of  the  equation.  Some  important  things  to  look 
for  are 

(1)  Symmetry. 

(2)  Points  where  the  locus  crosses  the  axes. 

(3)  What  values,  if   any,  of   one  variable  make   the  other 
imaginary. 

(4)  What   finite  values,  if   any,  of   one  variable  make  the 
other  infinite. 

(5)  How  increasing  or   decreasing   one  variable  will  affect 
the  other. 

(6)  What  value,  if  any,  does  one  variable  approach  when 
the  other  variable  becomes  infinite. 

50.    Illustrations.     Example  1.     To  plot  the  locus  of 

ar^-f.4/  =  16.  (1) 

If  the  equation   be  solved  for  x  and  y,  respectively,  there 
results 


x  =  ±  2^/^-f  (2) 

and  y=±  i  V16  -  af.  (3) 

(1)  Equation  (1)  shows  the  curve  to  be  symmetric  with 
respect  to  both  coordinate  axes  and  the  origin. 

(2)  If  y  =  0,  x=±4:',  if  x  =  0,  y  =  ±2.  Hence  the  curve 
meets  the  axes  at  (4,  0),  (-4,  0),  (0,  2),  and  (0,  -2). 

(3)  Equation  (2)  shows  that  if  2/^>4,  x  is  imaginary.  .*.  y 
cannot  be  greater  than  2  nor  less  than  —  2. 

Likewise,  eq.  (3)  shows  that  x  cannot  be  greater  than  4  nor 
less  than  —  4. 

(4)  No  finite  value  of  either  variable  can  make  the  other 
infinite. 

(5)  From  eq.  (3)  it  is  clear  that  as  x  increases  gradually 
from  0  to  4,  taking  all  values  in  that  interval,  the  value  of  y 
represented  by  the  positive  radical  steadily  decreases  from 
2  toO. 


LOCUS  OF  AN  EQUATION 


57 


(6)  Values  of  x  and  y  are  excluded  from  becoming  infinite 

by  (3). 

The  part  of  the  locus  that  lies  in  the  first  quadrant  may 
then  be  thought  of  as  generated  by  a  point  which,  starting  at 
(0,  2),  moves  gradually  to  the  right  and  downward,  until  it 
reaches  (4,  0).  A  few  additional  points  through  which  the 
curve  passes  will  then  suffice  for  a  fairly  accurate  drawing  of 
the  curve.  A  few  points  computed  from  eq.  (3)  are 
X  1  2  3  3.5, 
y    1.9     1.7    1.3       .96. 

The  curve  is  therefore  approximately  as  shown  in  Fig.  49. 

The  curve  is  an  ellipse,  as  will  be  shown  later. 


-Ip^^ 

^^                                                             ir- 

/                                                                               J^ 

^l 

^                                      0                                            ^^ 

^^                     y 

''^-^ .^---^ 

(1) 


Fig.  49. 

Example  2.    To  plot  the  locus  of 
a^-4.y^  =  l&. 

Solving  for  x  and  y,  respectively, 

aJ=±2V/T4,  (2) 

2/=±iVar^-16.  (3) 

(1)  Equation  (1)  shows  the  curve  to  be  symmetric  with  re- 
spect to  both  coordinate  axes  and  the  origin. 

(2)  If  a;  =  0,  2/  is  imaginary;  if  ?/=0,  x=  ±4.  Hence  the 
locus  does  not  meet  the  2/-axis,  and  meets  the  a;-axis  in  (4,  0) 
and  (-4,0). 

(3)  From  eq.  (2)  it  is  evident  that  x  is  real  for  all  real  values 


58 


ANALYTIC  GEOMETRY 


of  2/,  and  from  eq.  (3)  that  y  is  imaginary  for  all  values  of  A 
between  —  4  and  4,  and  is  real  for  all  other  values  of  x. 

(4)  No  finite  values  of  either  variable  makes  the  other  in- 
finite. 

(5)  Considering  the  value  of  y  corresponding  to  the  positive 
sign  of  the  radical  in  eq.  (3),  and  considering  positive  values 
of  X,  it  is  evident  that  as  x  increases  y  also  increases,  a  small 
change  in  x  making  a  small  change  in  y. 

(6)  As  X  increases  indefinitely,  y  also  increases  indefinitely. 
Moreover,  as  x  becomes  larger  and  larger,  Va?^  — 16  differs 

less  and  less  from  x.     This  may  be  proved  as  follows  : 

The  difference  between  x  and  Var^  — 16,  i.e.  x  —  ■\/x^  — 16, 
may  be  expressed  as 

(a;_Va^_16)(a;+Va^-16)_ 


Vaj2-16 


16 


-l-Var^-16 


01  ^^ 


without  limit  to  the  value  of  \ 
Now,  y  —  \x  is  easily  shown  to 


Now,  when  x  increases  indefinitely,  this  fraction  decreases  in- 
definitely and  approaches  the  limiting  value  0.     Therefore  as  x 

increases   indefinitely,    the    value 
of  y  approaches  nearer  and  nearer 

^x. 

—  1 

2 

be  the  equation  of  a  straight  line 
through  the  origin  and  the  point 
(2,  1).  Let  this  line  be  drawn. 
(Fig.  50.)  The  curve  will  then 
come  nearer  and  nearer  without  limit  to  this  line  as  x  becomes 
infinite. 

A  few  points  through  which  the  curve  passes  in  the  first 
quadrant  are 

iK    4     5       6       7       10, 
y    0     1.5     2.2     2.9      4.6. 

The  part  of  the  locus  which  lies  in  the  first  quadrant  may 
then  be  thought  of  as  generated  by  a  point  which,  starting  at 


Fig.  50. 


LOCUS  OF  AN  EQUATION 


59 


(4,  0),  gradually  rises  as  it  moves  to  the  right,  passes  through 
the  above  points,  and  approaches  nearer  and  nearer  to  the 
straight  line  whose  equation  is  y  =  ^  x.    (Fig.  50.) 

The  complete 
locus  is  obtained 
from  the  part  in 
the  first  quadrant 
by  considerations 
of  symmetry. 
(Fig.  51.) 

The  curve  is  an 
hyperbola,  as  will 
be  proved  later. 

The  straight 
line  to  which  the 

curve  approaches  indefinitely  near  as  the  point  generating  the 
curve  recedes  indefinitely  is  called  an  asymptote  of  the  curve. 

Example  3.     To  plot  the  locus  of 


Fig.  51. 


y 


Solving  for  x, 


(1) 

(2) 


3?/  +  l 
2,-2  ■ 

(1)  The  locus  is  not  symmetric  with  respect  to  either  coor- 
dinate axis  or  the  origin. 

(2)  If  05  =  0,  ?/  =  — ^  ;   if  ?/  =  0,  «  =  — i.     .*.  the  curve  meets 
the  axes  in  (0,  —  \)  and  {—\,^). 

(3)  No  real  values  of  either  variable  make  the  other  imagi- 
nary. 

(4)  If  X  =  3,  y  is  infinite ;  \i  y  =  2,  x  is  infinite. 

(5)  By  division,  eq.  (1)  may  be  written 

7 


2/  =  2  + 


(3) 


From  this  equation  it  follows  that  as  x  increases  from  a  nu- 
merically large  negative  number  to  3,  y  steadily  decreases  from 


60 


ANALYTIC  GEOMETRY 


a  value  a  little  less  than  2  to  —  oc.  As  a;,  increases  through  3, 
y  changes,  from  —  to  +,  and  as  x  increases  from  3,  y  steadily 
decreases  and  approaches  the  limiting  value  2  when  x  becomes 
infinite. 


Y 

"" 

p 

~ 

\ 

\ 

' 

\ 

\ 

k. 



"" 

^ 

"1 

^-^ 

0 

X 

\, 

^ 

L 

\ — 

_^ 

L 

u 

_ 

Fig.  52. 


4     6     10, 

9     ¥    3. 


The  following  points  are  on  the  locus : 
x     -5     -3     -2      0         1         2 

^  8"  6  5"  3^  2 

The  curve  may  then  be  sketched  as  in  Fig.  52.     The  lines 
a?  =  3  and  y  =  2  are  asymptotes  of  the  curve. 

Example  4.   To  plot  the  locus  of 

y  =  x{x  +  l){x^2). 

(1)  The  locus  is  not  symmetric  with  respect  to  either  co- 
ordinate axis  or  the  origin. 

(2)  The  locus  meets  the  axes  in  (0,  0),  (—  1,  0),  and  (  —  2,  0). 

(3)  No  real  values  of  either  variable  make  the  other  imagi- 
nary. 

(4)  No  finite  value  of  either  variable  makes  the  other  in- 
finite. 

(5)  Let  X  take  a  numerically  large  negative  value ;  then  y  is 
numerically  large,  but  negative.    As  x  increases  from  the  value 


LOCUS  OF  AN  EQUATION 


61 


assigned  toward  —  2,  each  of  the  factors  of  y  remains  negative, 
but  decreases  in  numerical  value;  y  therefore  remains  nega- 
tive, but  decreases  in  numerical  value  until  a;  =  —  2,  when  y  =  0. 
As  X  passes  through  the  value  —2,  the  factor  x-{-2  changes 
sign  and  becomes  positive,  the  other 
factors  of  y  remaining  negative  in 
sign  until  ic  =  —  1 ;  therefore  y  is 
positive  for  all  values  of -ic  between 

—  2  and  —  1.     As  a;  passes  through 

—  1, 2/  passes  through  0  and  remains 
negative  for  all  values  of  x  between 

—  1  and  0.  Asa;  increases  through 
0,  y  again  becomes  positive  and 
steadily  increases  as  x  increases  and 
becomes  infinite  when  x  becomes  in- 
finite. 

The  locus  may  then  be  generated 
by  a  point  which,   starting  indefi- 
nitely far  to  the  left  and  below  the 
origin,  steadily  rises  as  it  moves  to  the  right  until,  after  cross- 
ing the  X  axis  at  (  —  2,  0),  it  turns  at  some  value  of  x  between 

—  2  and  —  1,  descends  to  cross  the  a^-axis  at  (  —  1,0),  turns 
again  at  some  value  of  x  between  —  1  and  0  and  ascends  to 
cross  the  a>axis  at  (0,  0),  and  continually  thereafter  moves  to 
the  right  and  upward,  receding  indefinitely  from  both  axes. 

The  following  points  are  on  the  curve ; 


Y    / 
0  X 


Fig.  53. 


x 

-8 

-6 

-4 

-3 

-2 

-f 

y 

-336 

-120 

-24 

-6 

0 

|. 

X 

-1 

-  + 

0 

1 

2 

6, 

y 

0 

~l 

0 

6 

24 

336. 

The  curve  is  shown  in  Pig.  53. 
Example  5.     To  plot  the  locus  of 

f={x  +  2)(x-l)(x-S). 
(1)  The  locus  is  symmetric  with  respect  to  the  a^axis. 


62 


ANALYTIC  GEOMETRY 


(2)  The  locus  crosses  the  a^axis  at  (—2,  0),  (1,  0),  (3,  0), 
and  the  2/-axis  at  (0,  +  v'6)  and  (0,  —  VB). 

(3)  If  X  is  less  than  —2,  or  is  between  1  and  3,  y  is 
imaginary. 

(4)  No  finite  value  of  either  variable  makes  the  other 
infinite. 

(5)  Since  ^^  =  0  when  x=  —2  and  when  x=%  and  is  posi- 

tive for  all  values  of  x  between  —  2 
and  1,  therefore  as  x  increases  from 
—  2  to  1,  the  positive  value  of  y  must 
increase  from  0  when  x=  —  2  and 
then  decrease  to  0  when  x=\* 

As  X  increases  from  1  to  3,  y^  is 
negative ;  y  is  imaginary. 

As  X  increases  from  3,  y"^  becomes 
and  remains  positive  and  steadily  in- 
creases as  X  increases.  The  positive 
value  of  ?/)  therefore,  increases  as  x 
increases  from  3. 

(6)  When  x  becomes  infinite,  y  be- 
comes infinite. 

The  curve  then  consists  of  a  closed 
portion  between  x=  —2  and  ic  =  1,  and 
an  infinite  branch  to  the  right  of  a;  =  3. 
The  following  points  are  on  the  curve : 

a;_2-f-l         013       4        5         6         7        10, 

y       0  ±2.4  ±2.8  ±  2.5    0     0  ±5.5  ±7.5  ±11±14.7±27. 

The  curve  is  shown  in  Fig.  54. 

*  At  present  the  student  has  no  means  of  telling  that  y  does  not  change 
from  increasing  to  decreasing  and  from  decreasing  to  increasing  several 
times  as  x  increases  from  -  2  to  1  ;  nor  of  telling  where  a  change  of  this  kind 
takes  place.  The  investigation  of  such  questions  will  be  the  subject  oJ  a  later 
chapter. 


Y                    Z 

----------t-- 

-     7 

=g==t== 

X 

t  ^    , 

-----\--- 

fv — 

J- 

Fig.  54. 


LOCUS  OF  AN  EQUATION  63 


EXERCISE  XVII 

Discuss  the  following  equations  and  plot  the  curves  : 
1.    xy  =  4:.      2.    y  =  ix^.  3.   y-=—x.         4.  y  =  —x^. 

5.   y^  =  xK     6.   y=x(x-S).      7.  y  =  -^.     8.  ?/=(a;+5)x(x-3). 

9.   2/2  ^  4  a:^  =  4.      10.  y^  _  4  ^^2  _  4.        n.  ^2  ^  4  _  a;2, 

12.   y2^a;3.  13.  y  =  xK  14.    (x  + 2)  (y +  3)  =  1. 

15.   y  =  4x2  +  4.  16.    i  =  ?i^. 

y     a;-2 

17.   a:V  =  4.  18.  2/2=(a;-l)(ic-3)(ic-6). 

19.  y2=(a;-l)2(x-2).  20.  2/  ^ 


(x-l)(x-4) 


21    y= ^±2 .  22.  y=(^+2)(^-5) 

^      (a;_l)(x-3)  (x+l)(x-3) 

23.   y  =  g_(^  +  3)  (x  -  2)  ^  ^^    »u  =  a  constant. 

(x  +  l)Cx-4) 

25.  i)vi-2  =  6.  26.  v  =  32«. 

27.    s=16<2.  28.   y- 2  = 


x-3 

29.  A  light  is  placed  at  a  distance  h  ft.  above  a  plane  surface. 
Given  that  the  illumination  of  the  plane  at  any  point  varies  inversely 
as  the  square  of  the  distance  from  the  light,  and  directly  as  the  cosine  of 
the  angle  between  the  incident  rays  and  the  perpendicular  to  the  plane  ; 
prove  that  the  illumination  at  a  point  in  the  plane  at  a  distance  x  from 
the  foot  of  the  perpendicular  from  the  light  to  the  plane  is  given  by 

I  = — ,  where  O  is  a  constant. 

(X2  +  A2)f 

Plot  the  curves  for  A  =  20  ft.,  30  ft.,  and  40  ft. 


CHAPTER  V 


TRANSFORMATION   OF   COORDINATES 

51.  Change  of  axes.  The  coordinates  of  a  point  in  the  plane 
depend  upon  the  position  of  the  axes  to  which  the  coordinates 
are  referred. 

A  change  of  axes  will  change  the  coordinates.  The  equa- 
tions connecting  the  coordinates  of  any  point  in  the  plane  with 

the  coordinates  of  the  same  point 
when  referred  to  another  system 
will  next  be  derived  for  certain 
changes  of  axes. 

52.   Translation  of  axes.     Assume 

a  set  of  axes   OX  and   OY  and  a 

second  set  O'X'  and  O'Y'  parallel 

respectively  to  the  first  axes.     Let 

Pj^  55  0'  referred  to  OX  and  OF  be  (A,  A:).- 

Take  any  point  P  in  the  plane 

and  let  its  coordinates  referred  to  OX  and  OY  he  x  and  y,  and 

referred  to  O'X'  and  O'Y'  be  x'  and  y'.     Then  (see  Fig.  55), 

x  =  NP,     x'  =  N'F,     h  =  NN', 
y  =  MP,    y'  =  M'F,     k  =  MM', 

Kow  iVP=  ]^N'  4-  A^'P, 

and  MP  =  MM'  +  M'P, 

or  ,  a?  =  7*  +  a?'. 

y  =  k  +  y'. 

This  transformation  from  one  set  of  axes  to  the  other  is 
called  "  Translation  of  the  axes." 

64 


0 

Y 

N 

Y' 
N' 

X 

M 

0 

X' 

^ 

A' 

0' 

TRANSFORMATION  OF  COORDINATES 


65 


53  Rotation  of  axes.  Let  the  rectangular  axes  OX'  and  OY^ 
make  an  angle  ^  with  OX  and  Oy  respectively.  Let  any  point 
Phave  coordinates  (x,  y)  referred  to  OX  and  OY,  and  {x\  y') 
referred  to  OX'  and  OY'.  Let  OP=r  and  AX'OP  =  cf>'. 
ThenZXOP=cl>'  +  e. 
Then,  Fig.  56, 


Fig.  56. 

x'  =  r  cos  <^', 

y'  =  r  sin  <f>', 

x  =  r  cos  (<^'  +  0)f 

y  =  rsm(<f>'  +  e). 
Expanding  the  last  two  equations, 

x  =  r  cos  <^'  cos  ^  —  r  sin  <f>'  sin  0, 

y  =  r  cos  (f>'  sin  ^  +  r  sin  <f>'  cos  0, 
or  x  =  ie'  cos  6  —  y'  sin  0, 

y  =  x'  sin  6  +  y'  cos  9. 

These  equations  hold  for  any  point  in  the  plane.  They  ex- 
press X  and  y  in  terms  of  x'  and  y'. 

To  express  x'  and  y'  in  terms  of  x  and  ?^,  these  equations  may 
be  solved  for  x'  and  y',  or  the  equations  may  be  derived  as 
follows : 

InFig.  56,  let  ZXOP=<^, 
then  x  =  r  cos  <^, 

y  =  r  sin  <^, 


66  ANALYTIC  GEOMETRY 

a;'  =  r  cos  (<f>—  0)  =  r  cos  <^  cos  $  -\- 7-  sin  cj>  sin 6j 
y'  =  r  sin  (^<f>  —  6)  =  r  sin  <j>  cos  ^  —  ?•  cos  <^  sin  6, ' 
or  x'  =  x  cos  ^  +  2/  sin  ^, 

y^  =  y  cos  ^  —  a;  sin  ^. 

This  transformation  from  one  set  of  axes  to  the  other  is 
known  as  "  Rotation  of  the  axes." 

54.  Applications.  The  formulas  of  translation  and  rotation 
of  axes  may  be  used  to  simplify  equations,  thereby  making  the 
construction  and  classification  of  the  loci  easier. 

Example  1.     Consider  the  equation 

12  a;2  -  48  ic  +  3  ^2  +  6  2/  =  13. 

Let  the  axes  be  translated  to  a  new  origin  (Ji,  k),  the  formu- 
las for  which  are 

x  =  x'  +  li,      y  =  y'  -\-k. 

Substituting  these  values  in  the  equation,  it  becomes 
12  ic'2  +  3  2/'2  +  (24  7i  -  48)  oj' +  (6  A; -f  6)2/' + 12  7i2  +  3  A:2  _  48  ^ 

+  6A;-13  =  0. 

The  quantities  h  and  A;  may  have  any  real  values  assigned  to 
them.  If  they  be  so  chosen  that  the  terms  of  first  degree  in 
x'  and  y'  drop  out  of  the  equation,  the  equation  will  be  simpli- 
fied and  the  locus  will  be  symmetric  with  respect  to  the  axes 
O'X'  and  0'  Y'.     To  accomplish  this  it  is  only  necessary  to  let 

24/1-48  =  0, 
6  A;  +  6  =  0, 
from  which  7i  =  2,  fc  =  —  1. 

The  equation  then  becomes 

12  x"  +  Sy"  =  64.. 

This,  then,  is  the  equation  of  the  locus  referred  to  the  axes 
O'X' and  OT'. 


TRANSFORMATION  OF  COORDINATES 


67 


The  equation  is  now  easily  discussed  and  the  locus  plotted. 
Fig.  57  shows  the  locus  and  both  sets  of  axes. 

The  student  should  discuss  and 
plot  the  locus. 

Example  2. 


y'-Sy-\-4.x-h6 


0. 

the 


new 


Translate  the   axes   to 
origin  {h,  k)  by  means  of 

x  =  x'-{-h,y  =  y'-\-k. 
The  transformed  equation  is 
2/ '2  +  2  % '  +  A;2  -  8  /  -  8  A;  +  4  « ' 
+  4^  +  6  =  0. 

Here  it  is  not  possible  to  choose  h 
and  k  so  that  the  terms  of  first  de- 
gree in  x'  and  y'  will  drop  out,  since 
the  coefficient  of  x'  is  4.  They  can,  however,  be  so  chosen  that 
the  term  in  y'  and  the  constant  term  will  drop  out.  To  ac- 
complish this  it  is  only  necessary  to  let 

2k-S  =  0, 


Fig.  57. 


Y 

Y' 

X' 

/ 

0' 

X 

^^^ ' 

and  k'-Sk-\-4:h  +  6-. 

from  which    A*  =  4,  h  = 

The  equation  then 

=  0, 

=  #• 
be- 

comes 

y"-{-4.x'  = 

=  0. 

The  locus  is  now  easily 
constructed.     (Fig.  58.) 

Example  3. 
lU-+24a;?/-f42/2=20.    (1) 

In  equations  of  this 
form,  i.e.  equations  of 
second  degree  in  x  and  y  containing  a  term  in  the  product  xyj 
the  term  in  xy  may  be  made  to  drop  out  by  a  rotation  of  axes. 


Fig.  58. 


68 


ANALYTIC  GEOMETRY 


Let  x  =  x^  cos  6  —  y^  sin  0, 

y  =  ic'  sin  ^  +  y'  cos  6. 
Substituting  in  eq.  (1), 

11  (»'  cos  B  —  y'  sin  ey  +  21  {x^  cos  O  —  y'  sin  ^)  (x'  sin  $  ■\-y'  cos  ^) 
+  4  (a;'  sin  ^  +  ?/'  cos  ^)  ^  =  20.  (2) 

Expanding  and  collecting, 

^'-^=20.    (3) 


llcos^^ 
+24  cos  ^  sin  ^ 
+4sin2^ 


aj'2_22sin^cos^ 
+  24cos2^ 
-24sin2^ 
+  8  sin  0  cos  0 


a;y+llsin2^ 
-24  sin  ^  cos  (9 
+  4cos^^ 


It  is  now  possible  to  choose  9  so  that  the  coefficient  of  x'y^ 
will  become  zero ;  for  it  is  only  necessary  to  have 


or 
or 


24  (cos^  0  -  sin2  6)  =  14  sin  6  cos  By 
24cos2(9  =  7sin2^, 


tan  2  (9 


24 


(4) 


whose  tangent  is  -^y*- 


To  satisfy  eq.  (4),  let  2  ^  be  the  angle  in  the  first  quadrant 
Draw  the  right  triangle  with  sides  24 
and  7  as  in  Fig.  59.     The  hypotenuse  is  then  2^. 
.-.       sin2^  =  |i,  cos2^  =  -/5. 

Now  sin^  l9  =  1  (1  -  cos  2  B),  cos^  ^  =  i  (1  +  cos  2  ^), 
and 

sin  B  cos  ^  =  J  sin  2  ^. 

sin2<9=2^^,  cos2^  =  ||,  sin  ^  cos  ^  =  If. 
Substituting  these  values  in  eq.  (3)  and  dividing  the 
resulting  equation  through  by  125,  there  results 

4  a;'2  -  2/'2  =  4. 

Referred  to  the  new  axes  the  locus  is  much  more 
easily  constructed.  The  discussion  of  the  equation  is  very 
similar  to  that  of  example  2,  Art.  50. 


TRANSFORMATION  OF  COORDINATES 

Y 


69 


Fig.  60. 


The  locus  and  both  sets  of  axes  are  shown  in  Fig.  60. 
The  angle  through  which  the  axes  are  turned  is  tan~^  |. 


EXERCISE  XVIII 

Simplify  the  following  equations  by  a  translation  of  axes  to  remove 
the  terms  of  first  degree  where  possible,  and  by  a  rotation  of  axes  to  re- 
move the  terms  in  xy.     Plot  the  curves  and  all  coordinate  axes. 

1.  x2-6x+4?/2_^8?/  =  5. 

2.  4a:2-4?/2  +  a;-2?/  =  0. 

3.  4a;2  +  y2_i2ic  +  2y-2=0. 

4.  ic2  +  y-^-4a;  +  2y-ll  =  0. 

5.  ?/2-6  2/  +  8  =  4a;. 

6.  2a;2-62/2  4.xy-5a;  +  lly  =  3. 

7.  a;2-j-2a:2/  +  «/2-12x  +  2  2/  =  3. 

8.  a;2-a;?/-2«/2_x-4i/-2=0. 

9.  3x2  +  2a:y  +  3y2  =  8. 
10.  xy=4^. 


CHAPTER  VI 


THE    STRAIGHT   LINE 


55.   Theorem.     Every  straight  line   has  an  equation  of  first 
degree  in  Cartesian  coordinates. 
Two  cases  are  to  be  considered : 

(1)  The  line  parallel  to  a  coordinate  axis.     If  the  line  is 
parallel  to  the  a^axis,  then  all  points  of  the  line  have  equal 

ordinates.  .'.  ^  =  c,  where  c 
is  a  constant,  is  true  for  all 
points  of  the  line  and  for  no 
others.  It  is  therefore  the 
equation  of  the  line. 

Likewise,  a  line  parallel  to 
the  ,v-axis  has  an  equation  of 
the  form  x  —  c. 

(2)  The  line  not  parallel  to 
an  axis. 

Let  the  line  cut  the  2/-axis 
at  JV(0,  h). 

Let  P(x,  y)  be  any  point  of  the  line.     Through  P  draw  PM 
parallel  to  the  x-axis  to  meet  the  2/-axis  in  M.     Then  as   P 

moves  along  the  line,  the  ratio  —  will  remain  unchanged. 

For  if  P  is  any  other  point  of  the  line,  then,  by  similar  tri- 
angles, 

PM      P'M'' 

Let  this  constant  ratio  be  denoted  by  m. 
MN 


Fig.  61. 


Then 


PM 


=  m 


is  true  for  all  points  on  the  line,  and  for  no  others. 

70 


THE  STRAIGHT  LINE 


71 


From  the  figure  the  values  of  MN  and  PM  are  seen  to  be 
h  —  y  and  0  —  a;  respectively.     Therefore 

h 


y 


0 


or 


Fig.  62. 


This  is  therefore  the  equation  of 
the  line.  It  is  of  first  degree  in 
X  and  y. 

For  a  line  passing  through  the 
origin,  the  value  of  h  is  zero,  and 
the  equation  becomes 
y  =  mx. 

The  relation  between  the  lines 
y  =  mx  and  y=mx  +  b  is  shown 
in  Fig.  62. 

It  is  important  to  notice  that  if 

the  axes  are  rectangular,  the  constant  ratio  m,  or  — —,  is  the 

PM 

slope  of  the  line. 

56.  The  equation  of  first  degree.  Conversely,  every  equation 
of  first  degree  in  Cartesian  coordinates^  with  real  coefficients,  is 
the  equation  of  a  straight  line. 

The  general  equation  of  first  degree  is  of  the  form 

Ax  +  By-^C=0.  (1) 

Here  again  two  cases  are  to  be  considered : 
(1)  When  either  ^  or  5  is  zero.      Suppose  A  =  0.     Then 
B^O,  and  the  equation  may  be  written 

C 


y  = 


B 


This  equation  is  evidently  satisfied  by  all  points  on  a  line 
parallel  to  the  ic-axis,  and  by  no  others 


72  ANALYTIC  GEOMETRY 

Likewise,  if  B  =  0,  the  equation  represents  a  straight  line 
parallel  to  the  ?/-axis. 

(2)  When  neither  A  nor  B  is  zero.     Solve  the  equation  for 

''  A       O 

^  B        B 

Now  this  is  of  the  same  form  as 

y  =  mx  4-  b, 

which  was  found  in  the  preceding  article  to  be  the  equation 

of  a  straight  line,  and  since  a  straight  line  can  be  drawn  so 

that  m  and  b  will  have  any  assigned  real  values,  a  line  can  be 

C  A  AC 

drawn  so   that  b  = ,  and  m  =  — — .     Th.eny  —  —  —x——, 

B  B  B       B 

or  Ax  -\-By  +  0=0,  is  the  equation  of  this  line. 

Hence  Ax -\- By -\- C  =  0 

is  the  equation  of  a  straight  line. 

The  proofs  given  in  this  and  the  preceding  article  hold  for 
oblique  as  well  as  for  rectangular  coordinates.  It  is  only  in 
rectangular  coordinates,  however,  that  m  is  the  slope  of  the 
line. 

57.  The  conditions  which  determine  a  straight  line.     The 

position  of  a  straight  line  is  determined  when  there  are  known 
either, 

(1)  Two  points  on  the  line, 

(2)  A  point  on  the  line  and  the  direction  of  the  line, 

(3)  The  length  and  direction  of  a  perpendicular  from  a  fixed 
point  to  the  line. 

Considerations  of  these  conditions  lead  to  the  following 
special  forms  of  the  equation  of  the  straight  line. 

58.  The  two-point  equation.  Let  Pj  (.x'j,  y^)  and  Pg  (x2, 2/2)  be 
any  two  points.  To  find  the  equation  of  the  straight  line 
through  them. 


THE  STRAIGHT  LINE 


73 


Let  P{x,  y)  be  any  point  on  the  line.     Through  Pj  draw  a 
line  parallel  to  the  ^/-axis  to  meet  lines  parallel  to  the  ic-axis 
through  P  and  Pg  ill   ^  and 
Jfg  respectively.    Then  by  sim- 
ilar triangles, 

MP,^MF 
M^Pi     M2P2 

wnich  is  the  same  as 


y-vi 


X  —  Xi 


2/2  —  2/1        a?2  —  OCi 

This    equation     holds     for  Fig.  63. 

every  point  on  the   line,  and 
for  no  others.     It  is,  therefore,  the  equation  of  the  line. 

59.  The  intercept  equation.     In  the  last  article  let  the  two 
given  points  be  (a,  0)  and  (0,  6).     The  equation  then  becomes 

y  —  0 _x  —  a 
y  ^h      x  —  0 

Clearing   of  fractions,   transposing, 
and  dividing  by  a6,  the  equation  re- 
duces to        7:+^=l- 
a      o 

This  is  known  as  the  intercept  equa- 
tion of  the  line,  since  a  and  h  are  the 
Fig.  64.  intercepts  of  the  line  on  the  coordinate 

axes. 
In  this  equation  neither  a  nor  b  can  be  zero. 

60.  The  point-slope  equation.      Let  the  line  pass  through 
-fi(^i)  2/1)  and  have  a  slope  m.     Let  P(a',  y)  be  any  point  on  the 

line.     The  slope  of  the  line  joining  Pj  and  P  is  ^  ~  ^^ ,  which 


by  hypothesis  is  equal  to  m. 


X  —  Xi 


74 


ANALYTIC  GEOMETRY 


- — ^  =  m 

is  an  equation  which  holds  for  all  points  on  the  line  and  for  no 
others.     It  is,  therefore,  the  equation  of  the  line. 


Fig.  65. 

Clearing  of  fractions,  it  may  be  written 

y  —  Vi  =  m{x  —  a?i). 

This  equation  does  not  apply  to  a  straight  line  parallel  to 
the  2/-axis,  for  which  m  is  infinite. 

61.  The  slope  equation.  If  in  the  last  article  the  given  point 
is  (0,  6),  the  equation  reduces  to 

y  =  mx  -|-  6, 

which  is  the  slope  equation  already  considered.     (Art.  6^.) 

62.  The  normal  equation.  Let  the  distance  from  the  origin 
to  the  straight  line  be  p,  and  let  the  angle  which  this  perpen- 
dicular makes  with  the  a;-axis  be  a.  The  quantity  p  will  be 
considered  positive  always. 

Let  H  be  the  foot  of  the  perpendicular  from  the  origin  to 
the  line.  The  coordinates  of  H  are  then  p  cos  a  and  p  sin  a. 
The  slope  of  OH  is  tan  a.  Therefore  the  slope  of  the  given 
line  is  —  cot  a. 

Hence  the  line  passes  through  {p  cos  a,  p  sin  a)  and  has;  a 
slope  equal  to  —  cot  ol 


THE  STRAIGHT  LINE 

The  equation  of  the  line  is  therefore,  by  Art.  60, 
y  —p  sin  a  =  —  cot  a(x  —  p  cos  a). 

Y 


75 


Fig.  66. 


Replace  cot  a  by  -,  clear  of  fractions,  and  transpose ;  the 

sin  a 

equation  then  becomes 

X cos  a-\-y  sina—p  (cos^ a -\- sin^ a)  =  0, 
or  0?  COS  a  4-  y  sin  a  —  2>  =  0. 


63.  Reduction  of  Ax -{- By -{- C  =0  to  the  slope  intercept, 
and  normal  forms.  The  equation  Ax -{-  By -{-  C  =  0  may  be 
reduced  to  the  slope,  intercept,  and  normal  forms  as  follows : 

(a)  To  reduce  Ax -{- By -\-  C  =  0  to  the  slope  form.  Solve  the 
equation  for  y.     There  results 


A       C 

— x , 

B       B' 


y  = 


which  is  in  the  form  y  =  mx  -\-  b,  where  m  =  —  — ,  b 


C 
B'  b' 

The  method  fails  when  ^  =  0.  The  equation  cannot  then  be 
put  in  the  slope  form. 

(b)  To  reduce  Ax -\- By -\- C  =  0  to  the  intercept  form. 
Transpose  C  to  the  right-hand  side  of  the  equation,  and  divide 


76 


ANALYTIC  GEOMETRY 


by 


C;  the  resulting  equation  may  be  written 

X  .  y 


C 
A 


+ 


C' 
B 


1, 


7^ 


which  is  in  the  form  -  4-  "  =  1,  where  a  =  —  — ,  5  =  —  — . 
a     b       '  A'  B 

The  method  fails  if  either  A,  B,  or  C  is  zero.  If  (7  =  0,  both 
intercepts  are  zero.  If  either  A  or 
B  is  zero,  the  line  is  parallel  to  an 
axis  of  coordinates. 

(c)   To  reduce   Ax-\-By+C=0 
to  the  normal  form. 

X        Let    X  cos  a-\-y  sin  a  —  p  =  0    be 

the  normal  equation  of  the  line. 
The  foot  of  the  perpendicular  from 
the  origin  to  the  line  is  then 
(p  cos  a,  p  sin  a).  The  coordinates 
of  this  point  must  satisfy. the  equation  Ax-\-By-{-  (7=0. 

Ap  cos  a  +  Bp  sin  a  +  (7  =  0.  (1) 

Also  the  slope  of  the  perpendicular  is  the  negative  reciprocal 
of  the  slope  of  the  line ; 

J) 

tana  =  — .     [See  («)  of  this  article.]  (2) 

.JjL 


cos  a 


A 


±  Vl  +  tan^  a      ±  VA'  +  B^ 


Substituting  in  (2), 


sm  a  — 


Substituting  these  values  of  sin  a  and  cos  a  in  (1), 
-p  = 


C 


±  VA^  +  B' 


THE  STRAIGHT   LINE  77 

Substituting  these  values  of  sin  a,  cos  a,  and  p  in  the  normal 
equation  of  the  line,  there  results 


Hence,  the  equation  of  a  line  is  reduced  to  the  normal  form  by 
dividing  the  equation  of  the  line  through  by  the  square  root  of  the 
sum  of  the  squares  of  the  coefficients  of  x  and  y.  TJie  sign  of  the 
radical  should  be  taken  opposite  to  the  sign  of  C  so  that  p  will  be 
positive. 

64.  Illustration.  To  reduce  2a;  —  4?/  +  7  =  0  to  the  slope, 
intercept,  and  normal  forms. 

(a)  Solving  for  y  brings  the  equation  into  the  slope  form 

in  which  ^  =  i?  &  =  |. 

(&)  Transposing  the  constant  term,  7,  and  dividing  by  —  7, 
brings  the  equation  into  the  intercept  form 

-i   i 

in  which  a  =  — |,  &  =  J. 

(c)  Dividing  through  by  —  V2^  +  4^  brings  the  equation  into 
the  normal  form 

— -^  +  —^y F  =  ^> 

1.2  '^ 

in  which      cos  a  = ,  sin  «  =  -^ ,  p 


V5  V5  2V5 

65.   Applications  of  the  formulas.    By  the  use  of  the  formulas 

derived  in  this  chapter  the  equations  of  straight  lines  which 
satisfy  certain  conditions  can  be  easily  found. 

Illustrations. 

(a)  To  find  the  equation  of  a  straight  line  which  passes 
through  (3,  —  5)  and  makes  an  angle  of  30°  with  the  avaxis. 


78 


ANALYTIC  GEOMETRY 


The  slope  of  the  required  line  is  tan  30°,  or  — -.     By  sub- 

V3 
stituting  iu  the  equation  y  —  y^  =  m{x  —  x^  there  results 

2^  +  5  =  ^(^-3), 

which  reduces  to 

V3a;-32/  =  15-f  3V3, 
the  required  equation. 

(6)  To  find  the  equation  of  the  straight  line  which  passes 

through  (—3,  1)  and  makes 
an  angle  of  60°  with  the  line 
4a;-9y  =  12. 

Let    the    angles   which    the 
given  line  and  the  required  line 
make  with  the  a>axis  be  O-^  and 
6  respectively,  and  the  slopes  of 
these   lines   be   rrii  and    m   re- 
spectively.     Then  m^  =  tan  0^, 
m  =  tan   6.      But    mj  =  |^,   and 
^  =  ^,  +  60°.     (Fig.  68.) 
m  =  tan  ^  =  tan  {O^  +  60°) 
^  tan  ^1  + tan  60° 
1  -  tan  Oi  tan  60° 

-  I+V^  _4  +  9V3 
.      4V3      9-4V3 


^a 


Fig.  68. 


^144  +  97V3 
33 
Therefore  the  required  equation  is 
144  4- 97  V3 


2/-1 


33 


(a;  4- 3), 


or  approximately 


2/-l  =  9.450c  +  3). 


.  THE  STRAIGHT  LINE  79 

66.   The  point  of  intersection  of  two  straight  lines.     Let  the 

two  lines  whose  equations  are 

A,x-{-Biy+C,  =  0,  (1) 

and  A^  +  ^22/  +  C2  =  0,  (2) 

be  denoted  by  Li  and  L2  respectively.  In  eq.  (1)  x  and  y 
may  be  the  coordinates  of  any  point  on  L^,  and  in  eq.  (2)  x 
and  y  may  be  the  coordinates  of  any  point  on  L2,  and  hence  x 
and  2/  in  one  equation  are  not  the  same  in  general 
as  X  and  y  in  the  other.  If,  however,  the  lines 
intersect,  there  is  one  pair  of  values  of  x  and  y 
that  satisfy  both  equations;  namely,  the  coordi- 
nates of  the  point  of  intersection.  Conversely,  if 
values  of  x  and  y  can  be  found  which  satisfy  both 
equations,  they  are  the  coordinates  of  a  point  on 
both  lines,  i.e.  the  point  of  intersection.  Therefore,  to  find  the 
coordinates  of  the  point  of  intersection  of  two  lines,  solve  the 
equations  of  the  lines  as  simultaneous. 
What  if  the  lines  are  parallel  ? 

Illustration.     To  find  the  point  of  intersection  of 

Sx-\-2y  =  ll 
and  4:X  —  5y  =  T. 

Solving  the  equations  as  simultaneous,  the  values  of  x  and 
y  are  found  to  be  x  =  3,  y  =  l.  Therefore  the  point  of  inter- 
section is  (3,  1). 

Let  the  student  plot  the  lines  and  check  graphically. 

EXERCISE   XIX 

By  substituting  in  the  formulas  write  the  equations  of  the  straight  lines 
which  satisfy  the  following  given  conditions: 

1.  Passing  through  (2,  1)  with  slope  —  2. 

2.  Passing  through  (-  3,  7)  and  (2,  -  5). 

3.  With  X-  and  ^/-intercepts  3  and  -  8  respectively 


k 


80  ANALYTIC  GEOMETRY 

4.  With  2/-intercept  6  and  slope  2. 

5.  Passing  through  the  origin  with  slope  —  |. 

6.  With  a  =  SO''  and  p  =  4. 

7.  With  j9  =  5  and  m  =  —  |. 

8.  Passing  through  (2,  —  5)  parallel  to  3aj  —  2/4-4  =  0. 

9.  Passing  through  (0,  0)  perpendicular  to  ax  +  6y  +  c  =  0. 

10.  Passing  through  (a;i,  y{)  parallel  to  y  =  mx  +  &. 

11.  With  y-intercept  h  and  perpendicular  X,o  Ax-^  By  ■\-  C=0. 

12.  Passing  through  (/i,  A;)  parallel  to  iccos/S  +  ysin/3  =  q. 

13.  Passing  through  (e,  /)  parallel  to  Ix  +  my  +  n  =  0. 

14.  Passing  through  the  origin  and  perpendicular  to  gx  -\-fy  =  c. 

Reduce,  where  possible,  each  of  the  following  equations  of  straight 
lines  to  the  intercept,  slope,  and  normal  forms,  giving  the  values  of  a,  b, 
m,  a,  and  p. 

15.  Sx-4y  =  6.  17.   2x-6y  =  0.  19.    y-25  =  0. 

16.  y+2x  =  4.  18.    -x  +  2y  =  9. 

20.  Obtain  the  equation  of  the  straight  line  which  passes  through 
(1,  2)  and  makes  an  angle  of  60°  with  the  line  2x  —  6y  =  8. 

21.  Two  lines,  Li  and  X2,  intersect  in  (—3,  —  2)  ;  Li  has  a  y-inter- 
cept  equal  to  —  6  and  makes  tan-i  |  with  L2 ;  find  the  equations  of  the 
two  lines. 

22.  Find  the  equation  of  the  straight  line  of  slope  —  f  which  passes 
through  the  intersection  oi2y  --x  =  b  and  x-~Sy  =  1. 

23.  The  vertices  of  a  triangle  are  (1,  2),  (4,  —  6),  and  (—5,  —  3)  ; 
find  the  equations  of  its  sides. 

24.  Find  the  equations  of  the  perpendiculars  from  the  vertices  upon 
the  opposite  sides  of  the  triangle  in  example  23,  and  prove  that  they  meet 
in  a  common  point. 

25.  Find  the  equations  of  the  medians  of  the  triangle  in  exam,ple  23, 
and  prove  that  they  meet  in  a  common  point. 

26.  Find  the  equation  of  the  line  through  (h,  k)  making  tan-^m  with 
y  =  lx-[-b. 

27.  A  line  passes  through  (2,  5)  and  is  distant  3  from  the  origin ;  find 
its  equation.     How  many  solutions  ? 


THE  STRAIGHT   LINE  81 

28.  Show  that  Ax -\-  By  -\-  C  =  0 
and  Ax  +  By  +  K=  0 
are  parallel,  and  that 

Ax  +  By+C  =  0, 
and  Bx  —  Ay  +  K=0 

are  perpendicular. 

29.  What  set  of  lines  is  obtained  by  varying  b  in  the  equation 
y  =  mx  +  &  ?     What  set  of  lines  by  varying  m  ? 

30.  Discuss  the  effect  upon  the  line  Ax  -\-  By  +  C  =  0  of  changing 
each  of  the  constants,  keeping  the  other  two  unchanged. 

31.  Find  the  equation  of  a  straight  line  which  passes  through  the  inter- 
section of2x— y-\-5  =  0  and  a;  —  22/4-1=0,  and  makes  an  angle  of  45° 
with  y  =  2x. 

32.  Prove  that  ax  -\-  by  -}-  c  +k(Ix  -{-  my  -f-  w)  =  0  is  the  equation  of  a 
straight  line  which  passes  through  the  intersection  of  ax  -\-  by  -h  c  =  0  and 
Ix  +  my  4-  n=  0.     What  is  the  effect  on  the  line  of  varying  k  ? 

33.  Using  the  fact  expressed  in  example  32,  find  the  equation  of  the 
straight  line  which  passes  through  (3,  —1)  and  the  intersection  of 
2ic  +  4?/— 7  =0  and  7 x  —  2 y  -f- 13  =  0,  by  determining  the  proper  value 
of  A:. 

34.  Find  the  equation  of  the  straight  line  which  passes  through  the 
intersection  of  x  -\-Sy  —7  =0  and  y  —  Sx  =  2,  and  makes  an  angle  of 
135°  with  the  a;-axis. 

35.  Find  the  equation  of  the  straight  line  which  passes  through 
the  intersection  of  2x  ~  9y  =  18  and  7 y  +  bx  =  21,  and  is  parallel  to 
4x  +  6y  -S  =  0. 

36.  Find  the  equation  of  the  straight  line  perpendicular  to  Sy  =  7 x 
which  passes  through  the  intersection  of  x  -\-  2y  =  8  and  4 x  =  13 y. 

67.   Change    of    sign    of    AiK  -f-  Bj/  +  C.        The    expression 

Ax  -\-  By  +  C  is  j^ositive  for  all  points  on  one  side  of  the  line 
Ax-\-By-{-C  =  Oy  and  is  negative  for  all  points  on  the  other  side 
of  the  line. 

Proof.  I.  Srippose\B  ^  0.  The  line  is  then  not  parallel  to 
the  ^-axis.     Let  L  be  the  line  whose  equation  is  Ax-\-By+  (7=0. 


82 


ANALYTIC  GEOMETRY 


For  all  points  on  this  line 

^  B        B 

Let  (a?!,  2^1)  be  any  point  above  the  line.     Since  y^  is  greater 

than  the  ordinate  of  the  point 
on  the  line  with  abscissa  Xi,  it 
follows  that 

or     2/1  4-  -  a^i  +  -  >  0. 
^       B         B 

This  is  true,  then,  for   any 
point   above  the  line.      There- 
fore, for  all  points  above  the  line, 

Ax^  +  By^-\-C>0,iiB>0, 
and  Ax^  +  By^  +  C<0/\iB<0. 

In  either  case  the  expression  has  the  same  sign  for  all  points 
above  the  line. 

If  the  point  is  taken  below  the  line,  the  inequalities  are  all 
reversed.  Hence  the  expression  has  the  same  sign  for  all 
points  below  the  line,  but  that  sign  is  opposite  to  the  sign  of 
the  expression  for  points  above  the  line. 

II.  Suppose  ^  =  0.  Then  A^O.  The  expression  becomes 
Ax-\-C,  and  the  equation   of  the  line  becomes  Ax-^  C  =  0. 

C 

The  line  is  parallel  to  the  ?/-axis.     On  this  line  x  = To 

C 

the. left  of  the  line  aj< ,  and  to  the  right  of  the  line 


x> 


C 


A 

Therefore  Ax  +  C  has  the  same  sign  for  all  points 


on  one  side  of  the  line  and  has  the  opposite  sign  for  all  points 
on  the  opposite  side  of  the  line.  Hence  the  theorem  is  true 
in  all  cases. 


THE  STRAIGHT  LINE 


83 


An  important  special  case  of  this  theorem  is  the  following : 
The  sign  of  the  expression  Ax  -\-  By  -{-  G  is  the  same  as,  or 
opposite  to,  the  sign  of  C  according  as  the  point  (x,  y)  and  the 
origin  are  on  the  same,  or  opposite,  sides  of  the  line  Ax-\-By-{-  C=0. 
This  follows  at  once  from  the  theorem,  since  the  value  of  the 
expression  Ax -\-By-\-  Cis  C when  the  coordinates  of  the  origin 
are  substituted.  If  (7=0  and  A:^0,  the  student  can  easily 
show  that  the  sign  of  Ax  +  By  is  the  same  as,  or  opposite  to, 
the  sign  of  A,  according  as  (x,  y)  lies  to  the  right  or  left  of  the 
line  Ax  -\-By  =  0. 

68.   Illustration.     The  expression  3x-\-7 y  —  S  has  the  value 
2  at  (1,  1),  which  is  opposite  in  sign  to  C,  or  —  8.     Hence 
(1,  1)  and  the  origin  are  on  opposite  sides  of  the  line 
3x-\-7y-S  =  0. 

Also  the  expression  3x-\-7 y  —  8  has  the  value  —  1  at  (2,  ^), 
which  is  the  same  in  sign  as  —  8.  Hence  (2,  I)  and  the  origin 
are  on  the  same  side  of  the  line  3x-{-7y  —  S  =  0. 


69.  Distance  from  a  point  to  a  line.  A  numerical  example 
will  be  first  worked  through.  Let  it  be  required  to  find  the 
distance  from  (6,  —  3)  to  the 
line  3x  —  5y  =  7. 

Transform  to  parallel  axes 
through  the  given  point 
(6,  —3),  as  a  new  origin,  the 
equations  of  transformation 
for  which  are 

x  =  x'-{-6,  y  =  y'-3. 

Substituting   these   values   in 

the  equation  of    the    line,  it  Fig.  71. 

becomes 

3x'-5y'  +  26  =  0, 
which  is  the  equation  of  the  line  referred  to  the  new  axes. 


Y 

Y' 

^ 

0 

<■ 

X 

^ 

^ 

\ 

// 

- 

\ 

0' 

X' 

- 

(6, 

-3) 

84 


ANALYTIC  GEOMETRY 


The  distance  from  the  new  origin  to  the  line  is  given  by  the 
formula 

C 


P  = 


±^A'-irB' 


(Art.  63), 


which  here  becomes 


26 


p  =  — ::::::  =  4.46  nearly. 
V34 


70.   General  formula  for  the  distance  from  a  point  to  a  line. 

Let  the  given  point  be  Pq{xq,  y^,  and  the  given  line 

Ax^By-\-C=0. 

Transform  to   parallel  axes  through  Pq  as  a  new  origin,  for 
.  which  the  formulas  of  transformation 

are 

x==Xq-\-x\  y  =  yo-{-y'. 

x[    The  equation  of  the  line  referred   to 
the  new  axes  is  then 

A(x'  +  ^o)  +  B{y'  +  2/o)  +  O  =  0, 
or    Ax'-\-By' -{-Axo  +  Byo-^C=0. 

In  this  equation  x'  and  y'  are  the  vari- 
able   coordinates,   and    the    constant 
Fig.  72.  term  of  the  equation  is  Axq  -f  ByQ  4-  C. 

Therefore  the  distance,  d,  from  the  new  origin  to  the  line  is 
^  ^  Axq  +  Bpo  +  C 

This  distance  will  be  regarded  as  a  positive  quantity.  The 
sign  of  the  radical  must  therefore  be  taken  the  same  as  the 
sign  of  Axq  +  ByQ  +  O.  But  Axq  -\-  By^  -f  C  has  the  same  sign 
as  C  when  {x^^  y^)  and  the  origin  are  on  the  same  side  of  the 
line  Ax-{-By-\-C=0,  and  has  the  opposite  sign  to  C  when 
(xq,  2/o)  and  the  origin  are  on  opposite  sides  of  the  line  (Art.  67). 


THE  STRAIGHT  LINE  •       85, 

Therefore  the  sign  to  be  taken  with  the  radical  is  the  same  as 
the  sign  of  C  when  {xq,  y^  and  the  origin  are  on  the  same  side 
oi  Ax-\-By  -\-C=0,  and  opposite  to  the  sign  of  C  when  {xq,  y^ 
and  the  origin  are  on  opposite  sides  of  the  line. 

If  (7=0,  the  sign  to  be  taken  with  the  radical  is  the  same  as 
or  opposite  to  the  sign  of  A  according  as  {xq,  2/0)  lies  to  the 
right  or  left  of  the  line  Ax  -\-By  =  0. 

EXERCISE  XX 

1.  Find  the  distance  from  the  point  (3,  —  6)  to  the  line  7  x  —  6y  =  1S. 

2.  Find  the  distance  from  the  intersection  of  2x  —  9y  =  3  and 
—  by  —  4:X  =  12  to  x—b=6y. 

3.  The  vertices  of  a  triangle  are  ^(1,  4),  B(-  3,  -  5),  and  (7(6,  -  4); 
find  the  area  by  finding  the  lengths  of  AB  and  the  altitude  from 
C  to  AB.  Check  by  using  the  formula  for  the  area  of  a  triangle.  (See 
Art.  36.) 

4.  The  equations  of  the  sides  of  a  triangle  are  x-\-4y  —7  =  0, 
3x  +  y  +  l=0,  and  2y  —  6x-\-lS  =  0;  find  the  area  of  the  triangle  by 
finding  the  length  of  one  side  and  the  length  of  the  perpendicular  from 
the  opposite  vertex  to  that  side.  Check  by  using  the  formula  for  the 
area  of  a  triangle.     (See  Art.  36.) 

5.  Find  the  distance  from  the  intersection  of  2  x—  6y  =S  and 
8x  +y  +  13  =  0  to  the  line  through  (-  |,  4)  and  (0,  —  3). 

6.  Find  the  distance  from  (9,  —  1)  to  the  line  through  the  origin  with 
slope  —  ^. 

7.  Find  the  distance  from  (xi,  yi)  to  y  =  mx  +  6. 

8.  Find  the  distance  from  (xo,  2/o)  to  x  cos  a  -\- y  sin  a  =  p. 

9.  Find  the  equations  of  the  bisectors  of  the  angles  formed  by  the  two 
lines  2x  +  y  —  7  =0  and  4x  —  Sy  — 5  =  0.  Show  by  their  slopes  that 
the  bisectors  are  perpendicular  to  each  other. 

Suggestion.  The  distances  from  (xo,  yo)  to  2  x  +  y  —  7  =  0  and 
4iX  —  Sy  —  6  =  0  are,  respectively, 

2  Xq  -f  yo  -  7  ^^^  4  Xq  -  3  jj/o  -  5 
±\/5  ±V25 

Now  the  bisector  of  an  angle  is  the  locus  of  points  equidistant  from  the 
sides  of  the  angle.     Hence  to  obtain  the  equation  of  the  bisector,  place 


86  ANALYTIC  GEOMETRY 

the  above  values  for  the  distances  equal  to  each  other  and  remove  the 
subscripts  to  indicate  variable  coordinates.  The  proper  signs  of  the 
radicals  must  be  chosen,  as  explained  in  Art.  70. 

10.  The  three  sides  of  a  triangle  have  the  equations  3  ic  —  4  y  =  7, 
6x+12y  +  8  =0,  and  4x  +  3y— 12  =  0;  find  the  equations  of  the  three 
inner  bisectors  of  the  angles,  and  shov7  by  their  equations  that  they  meet 
in  a  point. 

11.  Find  the  equations  of  the  three  outer  bisectors  of  the  angles  of 
the  triangle  of  example  10,  and  prove  by  their  equations  that  two  of  the 
outer  bisectors  and  the  inner  bisector  of  the  remaining  angle  meet  in  a 
common  point. 

71.   Equations  of  the  straight  line  in  polar  coordinates. 
(i)  Equation  of  the  straight  line  through  two  points.     Let 

■f'lC*'!)  ^i)  ^nd  ^2  0*2)  ^2)  t)e  any  two  points  in  the  plane.  To 
find  the  equation  of  the  straight  line  passing  through  them. 
Let  P(r,  0)  be  a  point  on  the  line  as  shown  in  Eig.  73. 

Then  area  0PiP2  =  area  OPiP+area  OPP2. 

I.e.  I  i\r2  sin  (O^  —  ^1)  =  i  ^^1  sin  {6  —  0^)  +  ^  rr^,  sin  {B^  —  6), 

or         rjra  sin  (^o  —  ^1)  +  ^\^  sin  {Q  —  62)  +  rr^  sin  (d^  —  0)  =  0. 

This  equation  holds  for  any  position  of  P  on  the  line  be- 
tween Pi  and  Pg.  If  P  be  so  taken  that  Pg  lies  between  P  and 
Pi,  the  equation  that  holds  can  be  obtained  from  the  above 

equation  by  interchanging  r 
and  ?*2,  and  0  and  O2.  But  this 
interchange  only  changes  the 
sign  of  the  left  member  of  the 
equation,  and  since  the  right 
member  is  zero,  the  equation  it- 
self is  unchanged.  If  P  be  so 
taken  that  Pi  lies  between  P  and  Pg,  it  may  be  shown  in  the 
same  way  that  the  same  equation  holds.  Hence  the  above 
equation  holds  for  all  points  on  the  line,  and  clearly  for  no 
others,  and  is  tlieref  ore  the  equation  of  the  line. 


Fig.  73. 


THE  STRAIGHT   LINE  87 

The  same  equation  may  be  derived  at  once  by  equating  to 
zero  the  area  of  the  triangle  whose  vertices  are  P,  Pj,  and  Pg. 
(See  Art.  37.) 

(ii)  Equation  of  a  straight  line  in  terms  of  the  length  of  the 
perpendicular  from  the  origin  to  the  line  and  the  angle  which 
this  perpendicular  makes  with  the  initial  line. 

Let   the   perpendicular   from  the   origin  to   the  line   be  of 
length  py  and  make  an  angle  a  with 
the  initial   line.     Let  P(r,  ^)   be   a  ^ 

point  on   the  line.     Then  (Fig.   74)  /^S 

r  cos  (a  — 6)=  p.     Since   cos  (—A) c^:— lLu 

=  cos  A,  this  may  be  written 

r(ios{0-a)=p.  ^i«-'^4- 

The  student  should  show  that  this  equation  holds  for  all 
points  on  the  line. 

EXERCISE  XXI 

1.  Write  the  equation  of  the  line  through  (2,  30'')  and  (1,  60°). 

2.  Draw  the  line  whose  equation  is  r  cos  (0  —  60°)  =  5. 

3.  Draw  tlie  line  whose  equation  is  r  sin  ^  =  4. 

4.  Find  the  intersection  of  the  lines  r  cos  ^  =  8,  and  r  sin  ^  =  4. 

5.  Find  the  intersection  of  the  lines  r  cos  [^  —  sin-i  (f)]  =2,  and 
r  cos  [d  -  cos-i  ( j\)]  =  4. 

6.  Derive  the  equation  r  cos  (0  -  a)  =phy  substituting  in 

a;  cos  a  +  ?/  sin  a  =  p,  the  values  x  =  r  cos  6,  y  =  r  sin  d. 

7.  Derive  the  equation  of  the  straight  line  through  two  points  in  polar 
coordinates  by  substituting  in 

y  —  yi  _  a^  —  a:i 
y2-yi    xi-xx 
the  values  a:  =  r  cos  ^,  y  =  r  sin  ft 


CHAPTER   VII 

STANDARD  EQUATIONS  OF  SECOND  DEGREE 

CIRCLE,   PARABOLA,   ELLIPSE,   HYPERBOLA 

72.   The  circle.     The  equation  of   a  circle  of  radius  r  and 
center  at  (Ji,  k)  is 

Proof.     Denote  the  center  by  (7.     Let  P(x,  y)  be  any  point 

on  the  circle.  The  condition  that 
P  is  on  the  circle  is  expressed 
by  the  equation 

(7P  =  r. 


In  terms  of  the  coordinates  of  the 
points  it  becomes 

(x-hy^-{y-ky  =  r', 

P  (^.  y)     which  is  therefore  the  equation  of 
the  circle,   for  it  is  an   equation 
which  is  satisfied  by  all  points  on 
the  circle  and  by  no  others. 
If  the  center  is  at  the  origin,  the  equation  reduces  to 

73.   The  equation        ay^  +  y^  -\-  Doc  -{-  Ey  -[-  F  =  0.  (1 ) 

An  equation  of  this  form,  by  completing  the  square  in  the 
terms  containing  x  and  in  those  containing  y,  can  be  thrown 

88 


STANDARD  EQUATIONS  OF  SECOND   DEGREE      89 

into   the   form   of  the  equation  of   the   preceding  article   as 
follows: 

Add         ^J^^-Fto  both  sides  of  eq.  (1).     The  result  is 
4        4 

^  +  Dx  +  ^  +  f  +  Ey  +  ^  =  ^  +  ^-F, 
4  4        4        4 

x+fj+(.+g^^-^^-^^-     (2) 

Now,  if  D^  -\-  E^—4:  F  is  positive,  eq.  (2)  is,  by  the  preceding 
article,  the  equation  of  a  circle  with  center  at  [  —  — ,  —-^]  and 


radius  equal  to  ^VD^  ^E''-4.F. 

If  D^  +  E'^-4.F=0,  eq.  (2),  and  hence  eq.  (1),  is  satisfied 

bya;=— — ,  y=  — —  only ;  for  the  sum  of  two  terms,  neither 

of  which  is  negative,  can  vanish  only  when  the  terms  vanish 
separately. 

\iD'^-\-E'^-^F<^,  eq.  (2),  and  hence  eq.  (1),  is  satisfied 
by  no  real  values  of  x  and  y :  for  the  sum  of  two  quantities, 
neither  of  which  is  negative,  cannot  equal  a  negative  quantity. 

Hence  the  equation  . 

represents 

(1)  a  circle,  center  at  (-  ^,-  ^\  radius  =  \  VZ^4-^'-4i^, 

if  D2-f  J5;--4i^>0, 

(2)  apoint(^-|,-|^,ifi)^4--E^-4i^=0, 

(3)  no  locus,  if  Z)2  _|_  ^  _  4  2^  <  0. 

74.  The  equation  of  a  circle  through  three  points.  The 
equation  of  a  circle  through  three  given  points,  not  in   the 


90 


ANALYTIC  GEOMETRY 


same  straight  line,  can  be  found  by  use  of  the  equation  of  the 
preceding  article  as  is  illustrated  by  the  following  example : 

Example.     To  find  the  equation  of  a  circle  through  (2,  1), 
(-1,  3),  and- (-3,  -4). 

The  equation  of  the  required  circle  is  of  the  form 

^+f+Dx-^Ey+F=0.  (1) 

The  coordinates  of  each  of  the  given  points  must  satisfy  this 
equation.     Therefore 

5  +  2i)+  E  +  F=0, 
10-  D  +  SE-{-F=0, 
25-3D-4.E  +  F=0. 

The  values  of  B,  E,  and  F,  obtained  by  solving  these  equa- 


tions, are 


D 


Y 

^ 

^^. 

/ 

\ 

1 

\ 

X 

0 

\ 

\ 

/ 

V 

X 

/ 

<~- 

Fig   76. 

the  circle  with  center  and  radius  as  computed 


■VS   E  =  l,   F=-ii^. 

Substituting  these  values  in  eq.  (1) 
and  clearing  of  fractions,  the  re- 
quired equation  of  the  circle  is 

5(a.-2+  2/2)  -f  13  ic  +  7  2/  -  58  =  0. 

Using  the  formulas  of  Art.  73, 
the  center  and  radius  of  the  circle 
are  found  ta  be  (—1.3,  —.7)  and 
r  =  3.71.  .  .  . 

A  check  on  the  work  is  obtained 
by  plotting  the  points  and  drawing 
(Fig.  76.) 


EXERCISE   XXII 

Find  the  centers  and  radii  of  the  circles  represented  by  the  following 
equations.     Draw  the  figures. 

1.   a;24.y2^25.  2.  a;2  +  2/'^-4a;  +  6y  =  12. 

3.   a:2  +  2/2  +  8x-6y  =  0.  4.   2a;2  f  2  y2  _  7y  +  3a;  =  11. 

5.   (x  -  1)2 +  (y  + 2)2  =  0.  6.    (a;  - /i)2  +  (?/ -  A;)2  =  0. 

7.  m2  +  v2  4-  M  4- 1>  =  0.  8.   x2  +  y2  _  4  a;  +  6  y  +  14  =  0. 


STANDARD  EQUATIONS  OF  SECOND  DEGREE      91 

9.  x^-{-y^-ix-\-Qy-lS  =  0.      10.   x'^ -2ax  +  y^  =  0. 
11.  x^-2ax-\-y^  -2ay  =  0.  12.   x'^ -{- y^  -  ax  -  by  =  0. 

Find  the  equations  of  circles  which  fulfill  the  following  conditions : 

13.  Center  at  (-  1,  3),  radius  =  2. 

14.  Center  at  (a,  0) ,  radius  =  a. 

15.  Center  at  the  intersection  of  y  +  4ic+l  =  0  and  2x—y-\-6  =  0, 
and  passing  through  (2,  —  3). 

16.  Center  at  (2,  5)  and  tangent  to3x4-4y  =  ll. 

17.  Center  on  the  line  y  =  2x  and  passing  through  (0,  5)  and  (6,  1). 

18.  Passing  through  (0,  2),  (-  1,  3),  and  (5,  0). 

19.  Circumscribing  the  triangle   whose  sides  are  5x  +  Sy  —  IA  =  0, 
ix  —  Sy  +  6  =  0,  and  x  +  6  ?/  +8  =  0. 

20.  Inscribed    in   the    triangle   whose    sides    are    5x  +  l2y  —  2  =  0, 
4ic  +  3y  +  5  =  0,  and3x-4!/-15  =  0. 

21.  Tangent  internally  to  the  first  two  sides  of  the  triangle  mentioned 
in  example  20,  and  tangent  externally  to  the  other  side. 

22.  Prove  that  if  Pi(xi,  yi)  is  any  point  without  the  circle 

and  T  is  the  point  of  contact  of  a  tangent  drawn  from  Pi  to  the  circle, 
then,  P^2  =  (xi  -  hy  +  (yi  -  ky  -  r^. 

23.  Show  that  if  the  equation  of  the  circle  of  example  22  is 

x^  +  y^  +  Dx  +  Ey+  F=0, 
then,  KT^  =  xi^  +  yr^  +  Dxi  +  Eyi  +  F. 

24.  Prove  that  the  locus  of  points  from  which  equal  tangents  may  be 
drawn  to 

'  a;2  +  w2  -f  Dix  +  E^y  +  Fi  =  0, 
and  x'^  +  y^  +  D<ix  +  E<iy  +  Fg = 0, 

is  the  straight  line 

(l>i  -  D2)x  +  (^1  -  E^^y  -\-Fi-F2  =  0, 

or,  in  case  the  circles  intersect,  is  that  portion  of  the  line  not  inside  of  the 
circles. 
This  line  is  called  the  radical  axis  of  the  two  circles. 

25.  Show  that  if  two  circles  intersect,  their  radical  axis  passes  through 
their  points  of  intersection. 


92 


ANALYTIC  GEOMETRY 


26.  Find  the  equations  of  the  radical  axes  of  the  circles  of  examples  1, 2, 
and  3,  and  prove  that  they  meet  in  a  point. 

27.  Prove  that  the  three  radical  axes  of  any  three  circles  taken  in 
pairs  meet  in  a  common  point. 

28.  Prove  that  the  radical  axis  of  two  circles  is  perpendicular  to  their 
line  of  centers. 

75.  The  parabola.  The  parabola  is  the  locus  of  a  point 
which  moves  so  as  to  keep  equidistant  from  a  fixed  point  and 
a  fixed  straight  line. 

The  fixed  point  is  called  the  focus,  the  fixed  straight  line  the 
directrix,  of  the  parabola. 

To  obtain  the  equation  of  the  parabola,  let,  at  first,  the  direc- 
trix be  taken  as  the  axis  of  y  and  the  focus  at  the  point  (p,  0). 

Let  P(x,  y)  be  any  point  on  the  locus.  Join  P  and  F{p,  0), 
and  draw  MP  parallel  to  the  avaxis  to  meet  the  ^/-axis  in  M, 
Then  the  condition  that  P  is  the  point  on  the  locus  is  expressed 
by  the  equation 

FP  =  MP,  if  MP  is  positive,  and  by 
FP=-  MP,  if  MP  is  negative. 


P  ix,yy 


F(p,o) 


PCx,y) 


F(p,o) 


Fig.  77. 

Evidently  MP  is  positive  or  negative  according  as  (p,  0)  lies 
to  the  right  or  the  left  of  the  origin,  i.e.  according  as  p  is  posi- 
tive or  negative. 

Now  FP  =  -V(x-2)y-{-y^ 

and  MP  =  x. 

■y/{x-pY-\-y^  =  a;,  if  p  >  0,  (1) 

and  V(a;-p)2-f.2/2  =  -x,\ip<0.  (2) 


STANDARD  EQUATIONS   OF  SECOND  DEGREE      93 

Squaring  and  transposing,  either  of  these  equations  becomes 

f'  =  2px-p\  (3)* 

This  equation  may  be  written 


=  2,(.-|). 


Let  the  axes  be  translated  to  (-^,  0  j  as  a  new  origin.     The 

formulas  of  transformation  are 

a^  =  ^'  +  |,2/  =  y'- 

The  equation  of  the  parabola  referred  to  the  new  axes  is, 
therefore, 

y'2  =  2px\  -  (4) 

Dropping,  primes, 

y'  =  2poc  (5) 

is  therefore  the  equation  of  a  parabola  when  the  2/-axis  is  paral- 
lel to  the  directrix  through  a  point  halfway  between  the  focus 
and  directrix,  the  avaxis  passes  through  the  focus  and  is  per- 
pendicular to  the  directrix,  and  the  focus  is  at  ( ^,  0  ). 


It  is  important  to  note  that  in  eq.  (5)  the  abscissas  of  points 
on  the  parabola  vary  as  the  square  of  the  ordinates. 

76.   The  graph  of  2/2=2  i>ic. 

I.  p  positive. 

*  Equation  (3)  is  not  equivalent  to  both  eqs.  (1)  and  (2),  but  only  to 
(1)  if  p  is  positive,  and  to  (2)  if  p  is  negative.  For  on  retracing  the  steps 
from  (3)  the  eq.  \/(x  —  p)'^  -\-  y'^  =  ±x\s  obtained.  Now  the  +  sign  can 
only  be  used  when  x  is  positive,  since  the  radical  is  counted  positive.  But 
if  p  were  negative  when  x  is  positive,  then  \/(x  —  p)'^  +  y'^  would  be  greater 
than  X.  .'.  when  x  is  positive,  p  is  positive.  Therefore  the  +  sign  of  x  can 
be  taken  only  when  p  is  positive.  Hence  when  p  is  positive,  eq.  (3)  is 
satisfied  by  precisely  the  same  points  as  eq.  (1). 

In  the  same  way  it  can  be  shown  that  eq.  (3)  is  equivalent  to  eq.  (2) 
when  p  is  negative. 


94 


ANALYTIC  GEOMETRY 


(1)  The  curve  is  symmetric  with  respect  to  the  x-axis. 

(2)  When  x  =  0,y  =  0',  when  y  =  0,x  =  0.  The  curve  there- 
fore meets  the  axes  at  (0,  0)  only. 

(3)  All  negative  values  of  x  make  y  imaginary.  The  curve, 
therefore,  lies  to  the  right  of  the  ?/-axis. 

(4)  No  finite  value  of  either  variable  makes  the  other  infinite. 

(5)  As  X  increases,  the  positive  value  of  y  increases,  a  small 
change  in  x  making  a  small  change  in  y. 

(6)  When  x  becomes  infinite,  y  also  becomes  infinite. 

The  upper  half  of  the  curve  may,  therefore,  be  generated  by 
a  point  which,  starting  at  (0,  0),  moves  ever  to  the  right  and 
upward,  receding  indefinitely  from  both  axes. 

The  following  points  are  on  the  curve : 


xO      2     2 

8      2 


p      2p 


Sp 


4tp   Sp      50p    200p, 


y  0  ±^  ±p±pV2  ±2p  ±pV6  ±2pV2  ±4.p  ±10p  ±20p. 
The  curve  is  shown  in  Fig.  78  for  a  certain  value  of  p. 


F(|,o) 


p  positive 


Fig.  78. 


p  negative 


II.  p  negative.  This  case  differs  from  that  in  which  p  is 
positive  only  in  making  the  curve  lie  to  the  left  of  the  y-axis 
instead  of  the  right.     The  curve  is  shown  in  Fig.  78,  the  values 


STANDARD  EQUATIONS  OF  SECOND  DEGREE      95 

of  p  for  the  two  curves  being  numerically  equal,  but  opposite 
in  sign. 

77.  Axis  of  a  parabola.  Vertex.  The  straight  line  through 
the  focus  perpendicular  to  the  directrix  is  called  the  axis  of  the 
parabola. 

The  point  where  the  parabola  crosses  its  axis  is  called  the 
vertex. 

In  both  cases  of  Fig.  78  the  a>axis  is  the  axis  of  the  parabola, 
and  the  vertex  is  at  the  origin. 


78.   Parabola  with  axis  on  the  ^/-axis  and  vertex  at  the  origin. 

If  the  vertex  is  at  the  origin  and  the  focus  at  (0,  ^  j,  the  equa- 
tion can  evidently  be  obtained  from  that  of  Art.  76  by  ex- 
changing X  and  y.     The  equation  is  therefore 

The  two  cases  are  shown  in  Fig.  79. 


p  positive. 


a;2  =  2py. 
Fig.  79. 


p  negative. 


79.   The  arbitrary  constant  of  the  parabola 


96 


ANALYTIC  GEOMETRY 


Definition.  A  constant  which  may  have  any  vahie  in  an 
equation  is  called  an  arbitrary  constant,  or  a  parameter,  of  the 
equation. 

In  the  equation  of  the  parabola,  if  =  2})x,  since  p  may  have 
any  real  value,  it  is  an  arbitrary  constant  of  the  equation. 

Corresponding  to  each  value  of  p  there  is  a  definite  curve. 

The  curves  which  correspond  to  a  few  different  values  of  p 
are  sketched  in  Fig.  80. 


\ 

\ 

i  Y 

/ 

^ 

^ 

^     \ 

// 

/^ 

\  \ 

UP 

A  . 

< 

\ 

\      / 

/^ 

<■ 

\ 

\    / 

/ 

\ 

\j/ 

y 

P    =    "10 

p  = 

0 

'"' 

V    =  0 

X 

C 

-^^__ 

h 

1 

'\^ 

T 

\"^ 

/ 

\ 

\ 

7~ 

1     \ 

^. 

/    / 

\  ^ 

\_ 

/ 

7            f 

\ 

^V 

/ 

1 

\ 

y2  _  2  j3x. 
Fig.  80. 

80.   The  equations 

C2/'+/>a?+^2/  +  l^=0,   C^O,  JD^^, 

Equations  of  these  forms  can  by  a  translation  of  axes  be 
thrown  into  the  forms  x^  =  2py 

and  jf^^px 

respectively.  The  equations  therefore  represent  parabolas 
with  their  axes  parallel  respectively  to  the  y-axis  and  the 
o^axis. 


STANDARD  EQUATIONS  OF  SECOND  DEGREE      97 

A  numerical  example  will  make  this  clear. 

Example.  Sx^ -\-2x-\-5y -4.  =  0. 

Complete  the  square  in  the  terms  containing  x,  and  trans- 
pose the  other  terms  to  the  other  side  of  the  equation : 

or  (x  +  iy  =  -%(y-{i). 

Translate  the  axes  to  (— ^,  yf)  as  a  new  origin  by  means  of 
the  equations 

The  transformed  equation  is 


2/  =  2/'  +  tI- 


''^--f2/'. 


Y' 

0' 

Y 

-1                      X' 

/^ 

\:     > 

/ 

0         \ 

Fig.  81. 

This  is  the  equation  of  a  parabola  with  axis  on  the  new  y-axis, 
vertex  at  the  origin,  and  focus  at  (0,  —  -j^)  referred  to  the  new 
axes. 

Referred  to  the  old  axes,  the  vertex  is  (—  \,  |-f),  the  focus  is 
(— 1^,  T5^),  and  the  equation  of  the  axis  of  the  parabola  is 
x  =  -i. 

81.   The  equation      y  =  aoc^  +  6a?  +  c. 

This  equation  represents  a  parabola  with  its  axis  parallel  to 


98 


ANALYTIC  GEOMETRY 


the  ?/-axis.     For,  it  may  be  written 

\        a        4 ay  4a 

4  a  \       2aJ 


Let 


a^  + 


and  this  equation  becomes 


2a' 

6^  —  4  ac 


y\ 


which  is  the  equation  of  a  parabola  with  axis  on  the  new  i^-axis 
and  vertex  at  the  new  origin.     Hence,  referred  to  the  old  axes 

the  vertex  is  at  f , — — ^  ),  and  the  axis  of  the  pa- 

V-  2a'  4a     /  ^ 

rabola  is  parallel  to  the  2/-axis. 

The  parabola  extends  upward  or  downward  from  the  vertex 
according  as  a  is  positive  or  negative.  The  sign  of  6^  —  4  ac 
determines  whether  or  not  the  curve  crosses  the  a;-axis. 

Let  the  student  show  that  the  conditions  are  as  stated  in 
Fig.'  82. 


a  positive 
(1)    62_4ac>0. 


Fig.  82. 
y  =  a'3?'  -\- h% -^  c. 
(2)    62_4ac=0 


a  negative. 
(3)    62_4«c<0. 


STANDARD  EQUATIONS  OF  SECOND   DEGREE      99 

82.  The  parabolic  arch.  An  arch  of  height  ?i  and  span  2 1 
is  in  the  form  of  a  parabola  with  vertex  at  the  crown.  It  is 
desirable  to  compute  readily  the  heights  of  the  arch  at  vary- 
ing distances  from  the  center  of  the  span. 

Choose  the  axes  as  in  Fig.  83,  counting  y  as  positive  down- 


FiQ.  83. 


ward.     Since  in  a  parabola,  and  with  this  choice  of  axes,  the 
ordinate  varies  as  the  square  of  the  abscissa,  therefore 


This  form  of  the  equation  enables  one  to  compute  readily 
the  heights  at  varying  distances  from  the  center. 

E.g.  the  heights  at  distances  from  the  center  of  -,  -,  and 

3^  ^.     T     15h   3/i        ,  7h 

—  are  respectively  -——   — ,  and  ——  • 


EXERCISE  XXIII 

Plot  the  following  parabolas,  finding  the  vertex  of  each  and  reducing 
the  equation  to  the  standard  form.  In  each  case  compute  the  value  of 
the  discriminant,  b^  —  4  ac. 

1.   y  =  2x'^-Sx  +  5.  2.   y=-Sx^  +  4x-l. 

3.   y  =  x^-\-4:X  +  4.  4:.    5y-2x^+4x-3  =0. 

5.   2x^  +  Sx  +  y  =  4.  6.   Sy'^-2y  -\- 4x  -  S  =0. 


100 


ANALYTIC  GEOMETRY 


7.  Discuss  the  effect  upon  the  position  and  form  of  the  curve  caused 
by  separately  varying  the  quantities  a,  6,  and  c  in  the  equation 

y  =  ax^  -\-bx  +  c. 

8.  Discuss  the  equation  x  =  ay^  +  by  -\-  c.     (Compare  Art.  &1.) 

9.  A  parabolic  arch  of  60  ft.  span  is  20  ft.  high  at  the  center.     Com- 
pute the  heights  at  intervals  of  5  ft.  from  the  center. 

10.  Through  how  many  arbitrarily  assigned  points  can  a  parabola 
with  axis  parallel  to  one  of  the  coordinate  axes  be  passed  in  general  ? 
Name  some  exceptions. 

Find  the  equation  of  a  parabola  with  axis  parallel  to  the  ?/-axis  through 
the  three  points  (1,  0),  (3,  2),  and  (6,  8).     Draw  the  figure. 

11.  Find  the  equation  of  a  parabola  through  the  three  points  of 
example  10  with  axis  parallel  to  the  ic-axis. 

12.  Find  the  equation  of  a  parabola  through  the  points  (—  h,  yi), 
(0,  2/2),  and  (h,  yz)  with  axis  parallel  to  the  ?/-axls. 

13.  Find  the  equation  of  a  parabola  through  (1,  0),  (3,  2),  and  (6,  5). 

14.  Find  the  equation  of  a  parabola  with  axis  parallel  to  the  ?/-axis 
passing  through  (-  20,  0),  (0,  ^V),  and  (20,  0). 

Can  a  parabola  with  axis  parallel  to  the  aj-axis  be  passed  through  these 
points  ? 

15.  Show  that  any  line  parallel  to  the  axis  of  a  parabola  cuts  the 
parabola  in  one  and  only  one  point. 

83.  The  ellipse.  The  ellipse  is  the  locus  of  a  point  which 
moves  in  the  plane  so  that  the  sum  of  its  distances  from  two 

fixed  points  in  the  plane  is 
Y  constant. 

The  fixed  points  are  called 
the  foci. 

To  obtain  the  equation 
of  the  ellipse,  let  the  a;-axis 
be  taken  through  the  foci, 
and  the  origin  midway  be- 
tween the  foci.  Let  the 
distance  between  the  foci 
be  2  c.     The  foci -are  then 


P(:r,i/) 


Fig.  84. 


F(c,  0)  and  F\-  c,  0). 


STANDARD  EQUATIONS  OF  SECOND   DEGREE     101 

Call  the  given  constant  2  a,  where  2  a  >  2  c. 
Let  P{x,  y)  be  any  point  of  the  ellipse ;  then 

In  terms  of  the  coordinates  of  the  points,  this  becomes 


■^(^x  -  cf  +  f  4-  V  (a^  +  c)2  +  /  =  2  a.  (1) 

This  is  therefore  the  equation  of  the  ellipse. 

Equation  (1)  can  be  thrown  into  a  more  convenient  form 
free  from  radicals  as  follows  :  Transpose  the  second  radical  to 
the  right-hand  member  of  the  equation  and  square, 


a^_2ca;+c2+.y2=4a2-4aV(a;-f-c)2+2/2+a^+2ca;+c2+/.      (2) 
Canceling,  transposing,  and  dividing  by  4, 


cx  +  a^  =  aV(x  +  cy  +  yK  (3) 

Squaring, 

c  V  -f  2  a^cx  +  a*  =  a'x'  -f  2  a'cx  +  a  V  -f  ay.  (4) 

Canceling  and  collecting  terms, 

(a'  -(^)x'-\-  aY  =  a'  ip?  -  c^).  (5) 

Dividing  by  a^(a^  —  c^)j 

t^^£-  =  X.  (6) 

All  values  of  x  and  ?/  that  satisfy  eq.  (1)  also  satisfy  eq.  (6), 
but  in  obtaining  (6)  from  (1)  the  operation  of  squaring  was 
twice  performed,  and  in  this  process  there  are  introduced 
values  of  x  and  y  which  satisfy  eq.  (6)  but  do  not  satisfy 
eq.  (1).  However,  the  values  so  introduced  in  this  case  are 
imaginary,  and  hence  there  are  no  points  on  the  locus  of 
eq.  (6)  that  are  not  also  on  the  locus  of  eq.  (1).  For,  start- 
ing with  eq.  (6),  the  steps  may  be  retraced  until  eq.  (4)  is 
reached,  where,  upon  extracting  the  square  root,  a  double 
sign  is  introduced,  i.e. 

cx-\-a?=  ±  a^ix  +  cf  -{-  y^, 
or  —cx  =  a^  q:  aV(x-\-cy  +  y^  (3') 


102  ANALYTIC  GEOMETRY 

Multiply  by  4  and  add  (x  -\-  cy  +  y'^  to  both  sides, 


a^  +  2caj4-c2  4-?/2-4ca;  =  4a2:f  4 aV(a;  +  c)2 +  /  +  («;  + c)2  +  /, 
or  {x-cy-{-y'  =  (2aTV(x-\-cy-j-yy.  (2') 

Extract  the  square  root, 


±  V{x-  cy  -^  y'  =  2  aT  Vix-j-  cy  +  y% 

or  ±  V(«  -  cy  +  y'-  ±  V(a;  +  cy  +  y^  =  2a.  (1') 

Therefore,  if  (x,  y)  is  denoted  by  P, 

±PF±PF^  =  2a. 

Now  2  a  is  a  positive  quantity ;  hence  both  negative  signs 
cannot  be  used.  Also  FF'  =  2c.  The  difference  of  PF'  and 
PFis  therefore  less  than  2  c.  (Fig.  84.)  That  difference  can- 
not therefore  be  equal  to  2  a,  which  is  greater  than  2  c.  Hence 
the  only  allowable  combination  of  signs  for  real  values  of  x 
and  y  is  given  in 

+  PF-]-PF'  =  2a. 

Therefore  all  real  values  of  x  and  y  that  satisfy  eq.  (6)  also 
satisfy  eq.  (1).     Hence  eq.  (6)  is  the  equation  of  the  ellipse. 

Replacing  the  positive  quantity  a^  —  c^  in  eq.  (6)  by  6*,  the 
equation  becomes 

84.   Graph  of  ^'  +  ^'  =  1. 

a^     b^ 


Solving  for  y,  y  =  ±-  Va^  —  x^. 

a 

Solving  for  x,  a;  =  ±  -  V6^  —  y^. 

(1)  The  curve  is  symmetric  with  respect  to  both  coordinate 
axes,  and  the  origin. 

(2)  It  crosses  the  avaxis  at  (a,  0)  and  (—  a,  0)  and  the  y-axis 
at  (0,  b)  and  (0,  -  b). 


STANDARD  EQUATIONS  OF  SECOND  DEGREE     103 


(3)  If  X  is  less  than  —  a  or  greater  than  a,  y  is  imaginary. 
If  y  is  less  than  —  6  or  greater  than  h,  x  is  imaginary.  There- 
fore no  portion  of  the  locus  lies  to  the  left  of  x  =  —  a  ov  to  the 
right  of  x  =  a;  below  y  =  —  b,  or  above  y  =  b. 

(4)  No  finite  value  of  either  variable  makes  the  other 
infinite. 

(5)  In  the  first  quadrant,  as  x  increases  from  0  to  a,  y 
steadily  decreases  from  b  to  0. 

(6)  By  (3)  neither  variable  can  become  infinite. 
The  following  points  are  on  the  curve, 

a  a        3a      7^       9a 

4  2        T      T      To       ^ 


X     0 


y     b     .97  b     .87  6     Mb 
The  curve  is  sketched  in  Fig.  85. 


.48  6    .44  6     0 


Fig.  85. 

85.  Axes,  vertices,  center  of  the  ellipse. 

Definitions.  The  chord  of  the  ellipse  which  passes  through 
the  foci  is  called  the  major  axis  of  the  ellipse;  the  chord  at  right 
angles  to  the  major  axis  and  passing  through  its  center,  the 
minor  axis ;  their  intersection  the  center,  and  the  ends  of  the 
major  axis  the  vertices  of  the  ellipse. 

Thus  in  Fig.  85,  A' A  =  2  a  is  the  major  axis,  B'B  =  2  6  is  the 
minor  axis,  0  is  the  center,  A  (a,  0)  and  A' {— a,  0)  are  the 
vertices. 


104 


ANALYTIC  GEOMETRY 


In  terms   of    a   and   b   the   foci   are   F(Va^  —  b'-,   0)    and 


since 


—  ^2 


G^  —  &  or  c  =  V<: 


6-. 


86.   The  ellipse  with  major  axis  on  the  2/-axis.     In  Art.  83 
the  equation  —  -j-  ^  =  1  was  found  for  the  ellipse  with  center 

at  the  origin  and  with  major  axis  2  a  on  the  a;-axis.  If  the 
major  axis  2  a  were  taken  on  the  ?/-axis,  the  equation  would 
clearly  be  obtained  from  that  above  by  exchanging  x  and  y.. 
It  is  therefore, 

y"  ,y?  , 


^o  + 


1,  where  the  major  axis  is  2  a. 


If,  however,  the  major  axis  is  called  2  b  and  the  minor  axis 
2  a,  the  equation  becomes  the  same  as  that  of  Art.  83 ;  namely, 


-fi-' 


This  equation  therefore  represents  an  ellipse   with   major 
axis  on  the  a>axis  or  the  ^/-axis  according  as  a  is  greater  than 

Y 
Y 


a>  h 


Fig.  86. 


a  <b 


STANDARD  EQUATIONS  OF  SECOND  DEGREE      105 


or  less  than  b.     In  the  latter  case  the  foci  are  (0,   Vb^  —  a^) 
and  (0,  -  V6^^^-). 

87.  The  hyperbola.  An  hyperbola  is  the  locus  of  a  point 
•which  moves  in  the  plane  so  that  the  difference  of  its  distances 
from  two  fixed  points  of  the  plane  is  constant. 

To  find  the  equation  of  the  hyperbola,  as  in  Art.  83,  let  the 
fixed  points  be  F(c,  0)  and  F'  (—  c,  0),  and  let  the  constant  be 
2  a.  Here,  however,  2  a  <  2  c,  since  the  difference  between  two 
sides  of  a  triangle  is  less  than  the  other  side. 

Let  P{x,  y)  be  any  point  of  the  locus,  then  (Fig.  87)  either 

FP-F'P=2a  or  F'P-FP=2a, 
according  as  P  is  nearer  to  F^  or  F. 

FP-F'P=:±2a, 


P(=r,2/) 


Fig.  87. 

Expressed  in  terms  of  the  coordinates,  this  equation  becomes 

^{x-cf-^f--\/{x  +  cf^-y'=^±2a.  (1) 

This  is  therefore  the  equation  of  the  hyperbola. 

It  is  more  convenient  to  have  the  equation  free  from  radicals. 
Transposing  the  second  term  and  squaring, 

a^-2ca;+c2+2/-=a^+2ca;+c2f/±4aV(a;+c)2+2/^+4a2.      (2) 
Canceling,  transposing,  and  dividing  by  4, 

-  (ex  +  a^  =  ±  a^{x  +  cY-Jty\  (3) 


106  ANALYTIC  GEOMETRY 

Squaring, 

(fa^  +  2  a^cx  +  a*  =  aV  +  2  a^cx  +  aV  +  ay,  (4) 

Canceling  and  collecting  terms, 

(c2  -  a^  x^  -  ay  =  a\(?-a').  (5) 

Dividing  by  a^((?  —  a^, 
0?  v^ 


C^        (?  —   (^ 


1.  (6) 


This  is  precisely  the  same  form  as  eq.  (6)  in  Art.  83,  the  only 
difference  being  that  here  &  —  a^  is  positive,  whereas  there  it 
was  negative. 

Every  point  whose  coordinates  satisfy  eq.  (1)  also  satisfy 
eq.  (6).  That  conversely  all  points  which  satisfy  eq.  (6)  also 
satisfy  eq.  (1)  may  be  shown  as  in  Art.  83.  The  steps  by 
which  (6)  was  obtained  from  (1)  can  be  retraced,  but  a  double 
sign  must  be  used  when  the  square  root  is  extracted.  Hence, 
given  eq.  (6),  there  follows 


±V(a;-c)2  +  2/'±V(a;  +  c)2  +  /  =  ±2a.  (1') 

If  P{xj  y)  is*  any  point  on  the  locus  of  this  equation,  then 
±FP±F'P=±2a. 

The  same  sign  cannot  be  used  throughout,  since  the  sum  of 
two  sides  of  a  triangle  is  greater  than  the  third  side,  and 
2a<2c. 

The  same  signs  for  the  terms  on  the  left  and  the  opposite 
sign  on  the  right  cannot  be  used,  since  the  sum  of  two  positive 
quantities  is  positive. 

Hence  the  only  combinations  of  signs  left  is  that  where  the 

signs  of  the  terms  on  the  left  are  different,  which  is  equivalent 

to 

FP-rP=±2a. 


STANDARD  EQUATIONS  OF  SECOND   DEGREE     107 

Therefore  eq.  (6)  is  satisfied  by  only  those  points  which 
satisfy  eq.  (1).  Equation  (6)  is  therefore  the  equation  of  the 
hyperbola. 

Letting  the  positive  quantity  c^  —  a?  ■=  6^,  eq.  (6)  becomes 


88.   Graph  of  ^'-^'=1 


Solving  for  2^,  y=  ±  - Var'  —  a\ 

Solving  for  a;,  x=  ±-  ^/y^  +  b\ 

(1)  The  curve  is  symmetric  with  respect  to  both  coordinate 
axes  and  the  origin. 

(2)  It  intersects  the  avaxis  at  (a,  0)  and  (—a,  0),  but  does 
not  intersect  the  2/-axis. 

(3)  If  X  lies  between  —  a  and  a,  y  is  imaginary.     All  values 
of  y  make  x  real. 

(4)  No   finite   value   of    either   variable    makes   the   other 
infinite. 

(5)  In  the  first  quadrant  as  x,  starting   at  a,  increases,  y, 
starting  at  0,  steadily  increases. 

(6)  As  either  variable  becomes  infinite,  so  does  the  other. 
The  part  of  the  curve  that  lies  in  the  first  quadrant  may 

therefore  be  generated  by  a  point  which,  starting  at  (a,  0), 
moves  to  the  right  and  upward,  receding  indefinitely  from 
both  axes. 

The  following  points  are  on  the  curve : 


a 

3  a 
2 

2a 

Sa 

4a 

10  a 

100  a, 

0 

1.16 

1.7  b 

2.Sb 

3.9  b 

9.95  6 

99.99  6. 

108  ANALYTIC  GEOMETRY 

The  curve  is  shown  in  Fig.  88. 

The  foci  are  ( Va^  +  h\  0)  and  (-V«'  +  6',  0). 


89.  The  asymptotes  of  the  hyperbola.  By  observing  the 
table  of  values  of  x  and  y  in  the  preceding  article,  it  may  be 

seen  that  the  ratio  of  x  to  y  comes  nearer  and  nearer  to  -  as 

h 

X  increases. 

Consider  then  the  locus  of  the  equation  ^  =  - ,  or 

X      a 

b 
y==-  oc. 
a 

The  locus  of  this  equation  is  a  straight  line  with  slope  -  and 

(X 

?/-intercept  0  (Art.  55);  i.e.  it  is  a  straight  line  through  the 
origin  and  (a,  b). 

Let  yi  and  yj^  denote  the  ordinates  of  the  points  on  the  line 
and  the  hyperbola  respectively,  in  the  first  quadrant,  for  the 
same  value  of  x.     Form  the  difference 


2/i  -  2/a  =  -  (a^  -  Var^  -  a^), 


STANDARD  EQUATIONS  OF  SECOND  DEGREE     109 

which  may  be  written 


b{af-x''-\-a')  ah 


a  (x  H-  Vic^  —  a^)  x  +  V  a^  —  d^ 
from  which  it  is  evident  that,  in  the  first  quadrant,  yi  —  y^  is 
positive,  decreases  as  x  increases,  and  approaches  the  limiting 
value  0  as  a;  becomes  infinite.  The  curve  therefore  comes 
ever  nearer  to  the  straight  line  as  x  increases,  and  approaches 
indefinitely  near  as  x  becomes  infinite. 

The  line  y  =  -  a;  is  therefore  an  asymptote  of  the  curve. 

From  symmetry  the  same  line  is  an  asymptote  in  the  third 

quadrant,  and  y= a?  is  an  asymptote  in  the  second  and 

a 
fourth  quadrants. 

The  asymptotes  are  shown  in  Fig.  88. 

In  plotting  the  hyperbola  it  is  well  to  draw  the  asymptotes 
first.  They  will  serve  as  an  aid  in  sketching  the  curve  when 
a  very  few  points  have  been  located. 

90.  Axes,  vertices,  center,  of  the  hyperbola. 

Definitions.  The  points  of  intersection  of  the  hyperbola 
and  the  line  through  the  foci  are  called  the  vertices  of  the 
hyperbola;  the  line  joining  the  vertices  the  transverse  axis; 
the  middle  point  of  this  line  the  center  of  the  hyperbola ;  and 
the  line  through  the  center  perpendicular  to  the  transverse 
axis,  of  length  2Vc-  — a^,  the  conjugate  axis. 

Thus  in  Fig.  88,  A  and  J'  are  the  vertices,  A'A=  2  a  is  the 
transverse  axis,  the  origin  is  the  center,  and  B'B  =2  6  is  the 
conjugate  axis. 

91.  The  conjugate  hyperbola.  The  equation  of  an  hyper- 
bola with  foci  on  the  iz-axis  at  the  points  (0,  Va^  +  b^)  and 
(0,  —  Va^  4-  ^^)  and  transverse  axis  2  a  is  obtained  from  the 
equation  of  Art.  87  by  interchanging  x  and  y.  The  equation 
is  therefore  y^  _^'  —  i 

a'~b'~    ' 


no 


ANALYTIC  GEOMETRY 


If,  however,  the   transverse  axis  is  2  6  and  the  conjugate 
axis  2  a,  the  equation  is  ^ =  1,  or 


05^     y^ 


1. 


a' 


This  hyperbola  is  called  the  conjugate  of  ^  —  ^  =  1. 

ft  0 

It  is  easily  shown  that  the  conjugate  hyperbola  also  has  the 

lines  y  —  —  and  y= for  asymptotes.     The  proof  is  left 

ft  ft 

as  an  exercise. 

The  curve  is  shown  in  Fig.  89  together  with  the  hyperbola 


Fig.  89. 
Since  the  equations  of  the  asymptotes  can  be  combined  into 

the  one  equation  —  —  ^  =  0,  therefore  the  three  equations 


a' 


=  1, 


«  _  IT  _  _ 

a'      6^     ^' 


1, 


STANDARD  EQUATIONS  OF  SECOND  DEGREE     111 


represent  respectively  an  hyperbola,  the  conjugate  hyperbola, 
and  the  asymptotes. 

92.  The  equilateral  hyperbola.  If  6  =  a,  the  hyperbola  is 
called  the  equilateral,  or  rectangular,  hyperbola. 

The  equation  is 

05^  —  2/2  =  c^. 

The  asymptotes  are  y  —  nc  and  yzz^—ocj  and  are  therefore 
at  right  angles  to  each  other. 

93.  The  equilateral  hyperbola  referred  to  its  asymptotes  as 
axes.  In  the  equation  of  the  equilateral  hyperbola  of  the  pre- 
ceding article,  let  the  axes  be  rotated  through  an  angle  of 
—  45°.  The  asymptotes  then  become  the  axes.  The  formulas 
of  transformation  are 

«  =  ic'  cos  (  -  45°)  -  y'  sin  (  -  45°), 
2/  =  a;'  sin  (  -  45°)  +  y'  cos  (-  45°), 

V2 


Fig.  90. 


112  ANALYTIC   GEOMETRY 

Substituting  these  values  of  x  and  y  in  the  equation  of  tha 
equilateral  hyperbola 

there  results 

i{x"  +  2  x'y'  +  y")  -  i(a:'^-2  ^^  +  2/'^)  =a\ 

or,  dropping  primes,  xy  =  ~. 

This  is,  therefore,  the  equation  of  the  equilateral  hyperbola 
referred  to  the  asymptotes  as  axes. 

From  the  above  it  follows  that  if  two  variables  change  in 
such  a  way  that  their  product  remains  constant,  the  curve 
which  represents  the  equation  connecting  them  in  rectangular 
coordinates  is  an  equilateral  hyperbola.  E.g.  the  equation 
pv—Gi%  represented  by  an  equilateral  hyperbola. 

94.  The  equation  Aor  -\-  Ct/"' -{- nx  +  Ey  -{- F  =  0,  A  ^  0, 
C¥=0. 

An  equation  of  this  form  can,  by  a  translation  of  axes,  be 
transformed  into  one  in  which  the  terms  of  the  first  degree 
are  lacking.  For,  completing  the  squares  in  the  terms  con- 
taining X  and  in  those  containing  y,  the  equation  becomes 

Letting. =  .'-^^,  ,  =  y- A,  and£  +  ^-^=n 

the  equation  becomes 

Ax" -{- Cy"  =  F'. 

The  locus  of  this  equation  depends  upon  the  values  of  A,  C, 
and  F'. 

Suppose  I.     F'  =  0. 

(1)  Then,  if  A  and  G  are  of  the  same  sign,  no  real  values  of 
x'  and  y'  except  (0,  0)  will  satisfy  the  equation.  Hence  the 
locus  is  a  point. 


STANDARD  EQUATIONS  OF  SECOND   DEGREE     113 

(2)  If  A  and  C  are  opposite  in  sign,  Ax'^  +  Cy'^  =  0  can  be 
factored  into  two  factors  of  first  degree  in  x'  and  y',  and  there- 
fore the  locus  is  two  intersecting  straight  lines. 

11.   F=^0.     Divide  by  ii^', 


X 


'2  ^,12 


A  G 

(1)  If  —  and  —   are  both  positive,  the  locus  is  an  ellipse. 
(A  circle  itA=C.) 

77"  77'' 

(2)  If  —  and   _  are  both  negative,  no  real  values  of  x'  and 

A  C 

y'  satisfy  the  equation.     Hence  there  is  no  locus. 

77"  77'' 

(3)  If  —  and  —  are  opposite  in  sign,  the  locus  is  an  hyper- 

A  C 

bola. 

95.   Illustrations. 

Example  1.     3  x^  —  4:  y^  —  7  x -\-  5  y  -{-2  =  0. 
This  may  be  written 


H 

Ix 
3 

^36;^        V          4   ^64;      12 

25 
~16 

-2, 

or 

Z(x-lf-4.{y-iY  =  H. 

Let 

^=^'+hy  =  y'+h 

then 

3^r2_4y2^  25^ 

or 

^      1% 

This  is  the  equation  of  an  hyperbola  with  center  at  the  new 

origin,  transverse  axis  on  the  new  a;-axis,  with  a= A,  h  =— ^ 

12?  24 


114 


ANALYTIC  GEOMETRY 


Referred  to  the  old  axes  the  center  is  at  (J,  |).    (See  Fig.  91.) 
Y  Y' 


Fig.  91. 
Example  2.       x'-d  y"" +  7  x  +  9 y -^10  =  0. 
This  may  be  written 

^-\-7x  +  ^-9(f-y  +  i)  =  ^-i--10, 
or  (a,  +  i)2_9(2/-i)2  =  0. 

Let  X  =  X'-^,    y=,y'^l. 

Then,  x"-9y"=0, 

which  may  be  written 

{x'  +  3y')(x'-^Sy')  =  0. 
This  equation  is  satisfied  by  all  values  of  x'  and  y'  that 

Y'  Y 


Fig.  92. 


STANDARD  EQUATIONS  OF  SECOND  DEGREE     115 

make  either  x'  —  3  y'  =  0^  ov  x'  -^  S  y'  —  0,  and  by  no  other  values. 
The  locus  is  therefore  two  straight  lines  through  the  new  origin, 
with  slopes  J  and  —  -^  respectively. 

Referred  to  the  old  axes  the  point  of  intersection  of  the 
lines  is  (-  |,  i). 

96.  The  equation  Ax^  +  Bxy  -{.  Cy^  ^  Dx  -^  Ey  -\-  F  =0.  An 
equation  of  this  form  can  by  a  rotation  of  axes  be  reduced  to 
one  in  which  the  term  in  xy  is  lacking.  (Compare  Art.  54, 
example  3.)  The  resulting  equation  can  then  be  treated  as  in 
the  preceding  article. 

Usually,  where  the  xy-teTm  and  terms  of  the  first  degree 
appear  in  the  equation,  it  is  easier  to  first  remove  the  terms  of 
first  degree  by  a  translation  of  axes,  and  then  remove  the  term 
in  a^  by  a  rotation  of  axes.  It  is  not,  however,  always  possible 
to  remove  the  terms  of  first  degree. 

EXERCISE   XXIV 

Reduce,  where  possible,  the  following  equations  to  a  standard  form  of 
this  chapter.  Determine  the  axes,  position  of  centers,  vertices,  and  foci 
of  ellipses  and  hyperbolas ;  asymptotes  of  hyperbolas  ;  and  foci,  vertices, 
and  directrices  of  parabolas.     Sketch  the  curves. 

1.  (c2-6ic-4i/  +  l=0. 

2.  9x2  +  4y2_36a;-24y  +  36  =  0. 

3.  9x^-y^  +  S6x  +  2y  +  S6  =  0. 

4.  16a;2_y2_80a;-6i/  +  75  =  0. 

5.  3x^  +  y^  +  6x  +  7y-8  =  0.        7.  3a;2  +  4a;-y +  7  =  0. 

6.  5a;2_4^,2_^10a;-16?/  =  0.        8.   29x^ +  lQxy +  Uy^^4S  =  0. 
9.  21  a;2  +  52V2a;y- 68  2/2-324  =  0. 

10.  16a;2-24a;y  +  9y2_i80x+10y-75  =  0. 

11.  xy  =  S.  12.   xy-x2  =  5. 

13.  Sx^-^xy-\-6y^-{-6x-Sy  =  0. 

14.  14x2  +  45a;2/-142/2_i2a;  +  lly-2=0. 

15.  xy-\-2x  +  y+l=0. 


116  ANALYTIC  GEOMETRY 

16.  Prove  that  the  equation  of  a  parabola  with  vertex  at  (h,  k)  and 
axis  parallel  to  the  x-axis  is 

What  is  the  equation  if  the  vertex  is  at  (h,  k)  and  the  axis  is  parallel 
to  the  ?/-axis  ? 

17.  Prove  that  the  equations  of  an  ellipse  and  an  hyperbola  with  center 
at  (h,  k)  and  axes  parallel  to  the  coordinate  axes  are,  respectively, 


ni       ^        m       ~    ' 


62 

and  {x-hY_(y-lc)^^^^ 

18.  What  are  the  equations  of  the  asymptotes  of  the  hyperbola  in 
example  17  ? 

19.  Prove  that  xy  =  ax  +  hy  -\-  c  is  the  equation  of  an  equilateral  hyper- 
bola with  asymptotes  parallel  to  the  coordinate  axes,  if  —  c  =5^  ah.  By  a 
translation  of  axes  reduce  the  equation  to  the  form  xy  =  k. 

20.  What  are  the  equations  of  the  asymptotes  of  the  hyperbola  in 
example  19  ? 

21.  Prove  that  y  =  ^^         is  the  equation  of  an  equilateral  hyperbola, 

ex  +  d 

if  ad  =^  be,  and  that  the  asymptotes  are  x  = ,  y  =  -• 

c  G 

22.  In  the  equation  of  example  21  let  a  =  1,  b  =  2,d  =  S,  and  plot  the 
curves  for  the  following  values  of  c:  c  =2,  1.6,  1.4,  1,  .1.  Show  that  these 
curves  all  pass  through  the  same  two  points  on  the  axes. 

If  c  be  allowed  to  approach  the  limiting  value  0,  what  limiting  form 
does  the  hyperbola  approach  ?     What  limiting  form  if  c  approaches  f  ? 

23.  An  hyperbola  has  the  lines  x  =  2  and  y  =  4  as  asymptotes.  It 
passes  through  the  point  (4,  2) .     Find  its  equation. 


CHAPTER  VIII 

GRAPHS  OF  TRIGONOMETRIC,  EXPONENTIAL,  AND  LOGA^ 
RITHMIC  FUNCTIONS.  GRAPHS  IN  POLAR  COORDI- 
NATES 

97.  The  sine  curve.  Consider  the  graph  of  the  equation 
y  =  sin  X. 

Let  X  be  the  radian  measure  of  an  angle.  Let  x  and  y  be 
taken  as  rectangular  coordinates  of  points  in  the  plane,  the 
abscissa  of  any  such  point  being  the  number  of  radians  in 
the  angle,  and  the  ordinate  being  the  sine  of  that  angle. 
The  following  properties  of  the  locus  follow  from  the  prop- 
erties of  the  sine  of  the  angle. 

1.  The  locus  is  not  symmetric  with  respect  to  either  axis, 
but  is  symmetric  with  respect  to  the  origin,  since 

sin  (— ic)  =  —  sin  x. 

2.  The  locus  cuts  the  a;-axis  where  x  =  0,  it,  2 it,  •••;  —  tt, 
—  27r,  •••,  i.e.  where  x^kir,  k  being  any  positive  or  negative 
integer,  or  zero.     It  crosses  the  ?/-axis  only  at  the  origin. 

3.  All  real  values  of  x  make  y  real.  All  real  values  of  y 
between  and  including  —  1  and  1  make  x  real ;  all  other  values 
of  y  make  x  imaginary. 

4.  No  finite  values  of  either  variable  makes  the  other 
infinite. 

5.  As  X  increases  from  0  to  -.  y  increases  from  0  to  1, 

2 

As  X  increases  from  -  to  tt,  ?/  decreases  from  1  to  0, 

As  X  increases  from  tt  to  —,  y  decreases  from  0  to  —  1, 

117 


118 


ANALYTIC  GEOMETRY 


As  X  increases  from  -^  to  2  tt,  t/  increases  from  —  1  to  0. 

2 

6.  As  a;  increases  from  2  tt  to  4  tt,  2/  takes  in  succession  the 
same  set  of  values  that  it  takes  when  x  increases  from  0  to  2  tt. 
In  general,  since  sin  {A-\-2  hir)  —  sin  A,  where  k  is  any  positive 
or  negative  integer,  it  follows  that  if  x  is  increased  or  de- 
creased by  any  whole  multiple  of  2  7r,  sin  a;,  or  y,  is  unchanged. 
Hence  if  the  curve  be  plotted  through  an  interval  of  length 
27r  on  the  a;-axis,  other  portions,  of  the  curve  may  be  obtained 
from  this  portion  by  moving  it  to  the  right  or  left  through  the 
distance  2  tt,  4  tt,  6  tt,  etc. 

A  few  corresponding  values  are  shown  for  x  ranging  from  0 
to  TT,  and  the  curve  is  drawn  to  pass  through  the  points  so 
determined.  For  x  ranging  from  tt  to  2  tt  the  values  of  y  are 
those  given  below  changed  in  sign.     (Fig.  93.) 

IT  IT  TT  TT  2  TT  3  TT        5  TT 

6  4  32TX~6''^' 


0 


0 


1 

V2 

V3 

1 

V3 

V2 

1 

2 

2 

2 

2 

2 

2 

0. 


. 1 ^   MIU____  1   .__        _._■ -ft+ff-  — 

1    u  u^yjiil^-O'V  ^jLU^^^^^^^^^^^^^^^^ 

Ulf HI  MliiU 

:---::::  ::---i z^ J "s 3? 

::,.|=::  =  :=:i  =  :::::  =  :::v?  =  =  =  :  =  ::t:::::::^.|  =  :=::::==!:  =  :::=:  =  g  =  ::| 

;:^^^;=:"::=:"="::=;i^|===":!"|" 

^:::  :      ::      -z'::  :                                     '5              ::  +  :;?  ::::;:: 

--,           .               .                                             ,                     , 

::::::::  ^5; :::::;;?::::::::::::::::::::::::::::::::::  1:5  + ::::,^^f 

=*■■-= ^yTTri : 

:::  =  :::=:::4:t==±:=:::;,0.-1,^:  =  :::::::::  =  ::::=:==:==:::4=:-  =  --=  =  ::-- 

Sf=  SIN  a) 
Fig.  93. 


98.  Periodic  functions.  A  periodic  function  of  a  variable  is 
a  function  whose  value,  for  any  value  of  the  variable,  is  not 
changed  by  increasing  the  variable  by  a  definite  constant 
quantity.  The  least  positive  constant  quantity  by  which  the 
variable  can  be  increased  without  changing  the  value  of  the 
function  is  called  the  period  of  the  function. 


TRIGONOMETRIC  AND  EXPONENTIAL  FUNCTIONS    119 

Thus  sin  a;  is  a  periodic  function  of  x,  since  sin  (x-^^tt)  =sina;. 
Since  2  tt  is  the  least  positive  constant  value  by  which  x  may- 
be increased  without  changing  the  sine,  2  7r  is  the  period  of 
sinx. 

Again,  tan  ^  is  a  periodic  function  of  d,  since  tan(^+7r)=tan^. 
The  period  is  tt. 

Also  cos  {ax  +  6)  is  a  periodic  function  of  x  with  the  period 

,y  9 

— ,  since  increasing  xhj  ~  increases  ax  +  h  by  2  7r,  and  this 
a  a 

leaves  the  cosine  unchanged. 

99.  Graph  of  y  =  sin  (a?  +  a).  Let  a;'  =  a;  +  a,  or  x  =  x'  —  a. 
This  change  of  variable  means  geometrically  a  translation  of 
axes  to. the  new  origin  (—a,  0).  The  equation  referred  to  the 
new  axes  is  y'  =  sin  x'.  Figure  94  shows  the  curve  and  how  it 
is  located  with  respect  to  the  axes. 


■     >7;--->7-' -^— 1 

.                   -     -               _   ^^xt?     -                    3^  - .     -                   _ .     -         ,_ 

:::fs^:::::::::_::::^?^:.:::::j^:   ::::::::::::  S-: ::::::::::: ::-zh::: : 

±::::h;:::::::i::i??-:::   ::::::1:=:::::::::::::?^. -,"- 

:^^                 «;'               -             _t                                            *»               -^ 

1 

y=  8IN  (x+a) 
Fig.  94. 


100.   Graph  oi  y  =  sin  nx^  where  n  is  positive.     Let  ic'  =  nx, 

or  x  =  ~'     The  equation  then  becomes  y  =  sin x\  the  locus  of 
n 

which  is  shown  in  Fig.  93.     Now  the  substitution  of  a;  =  ^  can 

n 

be  interpreted  as  shortening  the  abscissas  of  all  points  in  the 

ratio  1 :  n  without  changing  the  ordinates.     If,  then,  the  curve 

2/  =  sin  a;  be  drawn,  the  curve  y  =  sin  nx  can  be  obtained  from  it 

by  shortening  the  abscissas  of  all  points  on  the  curve  y  =  sin  x 

in  the  ratio  1 :  n.     This  is  equivalent  to  compressing  uniformly 


120 


ANALYTIC  GEOMETRY 


in  the  direction  of  the  x-axis  any  portion  of  the  curve  y  ==  sin  st 

which  begins  at  the  origin  into  -  of  its  original  space,  the  end 

n 

of  the  curve  at  the  origin  to  remain  at  the  origin. 

It  is  also  equivalent  to  choosing  a  unit  on  tlie  ic-axis  equal  to 

-  of  the  unit  on  the  2/-axis  and  then  plotting  the  curve  y  —  sin  x. 
n. 

If  n  is  less  than  1,  the  contraction  of  the  curve  becomes  in 


fact  an  expansion. 


Graphs   of   1/  =  sin  3  a;   and   of  y  =  sin  (  — -  ]  are   shown   in 
rig.  95,  together  with  y  =  sin  x.  ^ 


2x 


-y-. 

gu::i-:=:=;=::::| 

■ 

0 

iTrNTmiTrrKM  ml 

l:l^lZ:t^^}i:V::-:-::: 

i-i==±^:l=±!: 

y—  SIN  X 


J  =  sm  3X 
Fig.  95. 


i;  =  siN^ 


m 


101.   Graph  of  y  =  sin  (nx  +  m).      Letting  x  =  x' ,  the 

n 

equation  becomes  y  =  sin  nx'.     Hence  translate  the  axes  to  the 

new  origin  ( ,  0  ]  and  construct  the  curve  y  =  sin  nx.     Com- 

\     n      J 

pare  Arts.  99  and  100. 

Figure  96   shows   the   locus   of  ?/  =  sin  (nx  +  m)  for  n  =  2, 

m=  —  1. 


Fig.  96. 


TRIGONOMETRIC  AND  EXPONENTIAL  FUNCTIONS   121 


102.  Graph  of  y  =p8m(nx-\-rn).  The  graph  of  this 
equation  can  be  obtained  by  multiplying  each  ordinate  of 
y  =  sin  {nx  +  m)  by  p.  The  curve  is  shown  in  Fig.  97  for  p  and 
71  both  positive. 


y^p  sinC«xh-w) 
Fig.  97. 

103.   The  exponential  curve,  y  =  a'',  where  a  is  positive. 
(Only  positive  values  of  y  are  considered.) 

(1)  The  locus  is  not  symmetric  with  respect  to  either  coordi- 
nate axis  or  the  origin. 

(2)  It  intersects  the  ?/-axis  at  (0,  1),  but  does  not  meet  the 
a;-axis. 

(3)  For  every  real  value  of  x  there  is  one  real  and  positive 
value  of  y.     Only  this  value  is  considered. 

(4)  No  finite  value  of  x  makes  y  infinite. 

(5)  lia  <  l,y  approaches  zero  as  ar becomes  infinite  positively. 
If  a  >  1,  i^  approaches  zero  as  x  becomes  infinite  negatively. 

(6)  If  a  >  1,  2/  increases  always  as  x  increases. 
If  a  <  1,  y  decreases  always  as  x  increases. 

If  a  =  1,  the  curve  becomes  the  straight  line  y  =  1. 
Figure  98  shows  a  few  curves  whose  equations  are  of  the 
form  y  =  a'',  for  certain  values  of  a. 

Values  of  y  may  be  computed  by  logarithms. 
E.gAi  a  =  e  =  2.718  •••* 


*  The  quantity  1  + 


4-+- 


+^- 


\1  '  \1'  li^."*"!!    \1 

by  e  and  is  called  the  natural  base  of  logarithms, 
in  more  advanced  mathematical  work. 


—  =  2.71828  —  is  denoted 
It  is  of  much  importance 


122 
then 


ANALYTIC  GEOMETRY 

Iogio2/  =  a5logio2.718  ... 
=  .4343  a;. 


Y 

r 

1 

1 

1 

/ 
1 

/ 

u 

o 

t 

11 

ij 

? 

\ 

/ 

\ 

1 

/ 

\ 

1 
1 

\ 

, 

1  / 

t 



0^ 

\__ 

Ur 

jy 

^ 

_ 

0 

X 

Fig,  98. 

The  following  points  are  on  the  curve  y  —  e*, 

a;     _5     _3     _2     -1     0       1       2      3 
y    .007      .05      .14      .37     1     2.7     7.4     20 


5, 
148. 


After  a  few  points  on  the  curve  have  been  obtained,  other 
points  are  easily  found  by  noticing  that  when  x  is  doubled, 
y  is  squared ;  when  x  is  tripled,  y  is  cubed ;  etc.  This  follows 
at  once  from  the  law  of  exponents,  a"*  =  (cu^y. 

104.  The  logarithmic  curve,  y  =  \o^a^.  This  curve  is  the 
same  as  that  oi  y  =  a'  with  x  and  y  interchanged.  The  curves 
for  y  =  logio  X  and  y  =  logg  x  are  shown  in  Fig.  99. 


TRIGONOMETRIC  AND  EXPONENTIAL  FUNCTIONS    123 


Since  log^  x  =  log^  x  log^  a,  when  the  curve  y  =  log^  a;  has 
been  constructed  for 
any  value  of  a,  the 
curve  y  =  logf,  x  can 
be  easily  obtained  from 
it  by  multiplying  all 
the  ordinates  of  the 
first  curve  by  logj  a. 

E.g.  the  ordinates  of 
y  =  log,x  are  2.3026 
times  the  correspond- 
ing ordinates  oi  y  =  logi6  a;,  since 

log.  10=     ^ 


1 


logio  e     .4343 


Fig.  99. 


=  2.3026. 


105.   Graph  of  y=e ""%  where 


Y                                I] 

i_ 

r 

I 

I 

3 

t 

1 

\ 

V 

5 

V 

\ 

\ 

^^^ 

""^ Z±I     X 

1                              0 

L 

e  =  2.718  •••  and  a  is  positive. 
Since    e""*  =  (e-")'  — 

this  curve  is  of  the 


©■ 


form 


Fig.  100. 


same  lorm  as  y  =  a% 
where  a^  is  less  than  1. 
The  curve  is  therefore 
as  shown  in  Eig.  100. 
(Compare  Art.  103.) 

The  rapidity  with 
which  the  curve  falls 
as  the  tracing  point 
moves  from  left  to  right 
depends  upon  the  value 
of  a. 

106.     Graph      of 

y  =  he~'^  sin  {nx  +  m). 
This  graph  is  easily  ob- 
tained by   plotting  sep- 


124 


ANALYTIC  GEOMETRY 


arately  the  graphs  of  y  =  e~"*  and  y  =  b  sin  (nx  4-  m)  and  multi 
plying  together  the  corresponding  ordinates.  The  form  of  the 
curve  is  shown  in  Fig.  101.  This  is  an  important  curve  in  the 
theory  of  alternating  currents. 


y  =  b  e~^^  SIN  {nx+m)y  where   a=  —.4,    6=1.5,    ii  =2,    m=  ~3. 
Fig.  101. 


EXERCISE   XXV 

Plot  the  following  curves.     (The  letters  i,  q,  t.  are  variables.) 

1.  y  =  cos  X. 

2.  y  =  tan  x.    (Divide  ordinates  of  sine  curve  by  those  of  cosine  curve. 

3.  y  =  CSC  X.     (Obtain  from  sine  curve.) 

4.  y  =  sec  X. 

5.  y  =  -^^  •     (Examine  carefully  near  x  =  0.) 

sin  ic 


S.   y  =  cos  2  X. 
8.  y  =  tan  2  x. 


7.   2/  =  cos3x. 
d.   y  =  tan  3  x. 


TRIGONOMETRIC  AND  EXPONENTIAL  FUNCTIONS     125 

10.   y  =  sin  (3x— 1).  11.    y  =  cos (x -\- a) . 

12.   y  =  cos  (nx).  13.   y  =cos(nx-\-m). 

14.  Show  that  the  graph  ot  y  =  cos  x  is  the  same  as   the   graph  of 

y  =  sin  a;  moved  parallel  to  the  ic-axis  the  distance   -  in  the  negative 
direction. 

15.  By  what  change  in  position  can  the  graph  of  y  =  cot  x  be  made  to 
coincide  with  the  graph  of  y  =  tan  x  ? 

16.  I  =  6e-«'. 

17.  i  =  6(1  —  e"«') .     (Combine  the  graphs  of  i  =  b  and  i  =  &e-«'.) 


18. 

i  =  bte-"^. 

19. 

gz=6  +  c(l  + A;Oe-«\ 

20. 

q  =  asinnt  +  b  sin 

3n^ 

21. 

y  =x  +  &mx. 

22. 
24. 

y  =  sin  a;  +  cos  x. 
y  =  sin2  X. 

23. 

—a)- 

26. 

y  =  sinii  X. 

25. 

?/  =  sin^  X. 

28. 

y  =  smx-\-  sin  2  x. 

27. 

yii  =  sin  X. 

30. 

y^^  =  sin  X. 

29. 

y  =  sini*^  X. 

32. 

y  =  sin  a;  +  sin  3  X  +  sin  5  x. 

31. 

y  =  sin"  X. 

34. 

q  =  e-'  sin  2  «. 

33. 

?/ =  e-2a:sjnjc^    - 

36. 

y  =  sin-ix. 

35. 

I  =  e ""'  sin  w^. 

38. 

y^sin-ix. 
cos-ix 

37. 

y  =  tan~i  x. 

39. 

y  =  sin  X  -f  ^  sin  3  X  + 

^sin5x  +  |sin7x. 

107.  Plotting  in  polar  coordinates.  The  methods  used  in 
plotting  a  curve  in  polar  coordinates  do  not  differ  essentially  from 
those  used  in  plotting  curves  in  rectangular  coordinates.  The 
difference  comes  mainly  in  the  manner  of  locating  the  points. 
The  following  examples  will  sufficiently  illustrate  the  methods. 

Example  1.  To  plot  in  polar  coordinates  the  curve  whose 
equation  is  r  =  a  cos  6. 

The  following  pairs  of  values  of  r  and  6  may  be  at  once 
written,  using  approximate  values  of  r : 

6  0°       30°       45°  60°    90°       120°         135°         150°  180° 

r  a    .S7  a     .71a  .5  a       0     —.5  a     —.71a     —.87a  —a 

0         210°          225°  240°     270°     300°      315°     330°  360° 

r  —.87a     —.71a  —.5  a          0       .5  a     .71a     .87  a  a 


126 


ANALYTIC  GEOMETRY 


An  examination  of  the  variation  in  r  as  6  increases  from  6 
to  360°  shows  that  as  6  increases  from  0  to  90°,  r  decreases 
from  a  to  0 ;  as  ^  increases  from  90°  to  180°,  r  is  negative  and 
decreases  from  0  to— a,  the  point  (r,  B)  tracing  out  apart  of  the 
curve  in  the  fourth  quadrant ;  as  $  increases  from  180°  to  270°, 
r  remains  negative  and  increases  from  — a  to  0,  the  point  {r,  6) 
tracing  over  again  the  part  of  the  curve  already  traced  in  the 
first  quadrant;  as  9  increases  from  270°  to  360°,  r  increases 

from    0   to    a,    the    point 
p  (r,  6)  tracing  over  again 

the  part  of  the  curve 
already  traced  in  the 
fourth  quadrant. 

If     6    is      allowed     to 

increase   beyond    360°  or 

to   take    negative   values, 

P  cos  0      takes      on      the 

r  =  a  cos  e.  same    series     of     values 

Fig.  102.  already     obtained,     since 

cos  {6  ±  360°)  =  cos  0,  and  no  new  points  are  obtained.     The 

curve  is  therefore  as  represented  in  Eig.  102. 

The  curve  appears  to  be  a  circle.  That  it  is  so  in  fact  may 
be  proved  as  follows :  Take  any  point  P{r,  0)  on  the  curve ; 

then  r  =  a  cos  0  or  cos  $  =  —     Therefore  Z  OPA  must  be   a 

a 

right  angle.     Therefore  the  curve  is  a  circle. 


Example  2.  To  plot  in  polar  coordinates  the  curve  whose 
equation  is  r^  =  a^  cos  3  6. 

For  ev^ry  value  of  0  which  makes  cos  SB  positive  there  are 
two  values  of  r  which  differ  only  in  sign,  and  for  every  value 
of  B  which  makes  cos  3  B  negative  the  values  of  r  are  imaginary. 
In  order  then  for  r  to  be  real,  3  B  must  be  an  angle  in  the  first 
or  fourth  quadrant. 

Let  the  positive  value  of  r  be  taken  for   discussion  first. 


TRIGONOMETRIC  AND  EXPONENTIAL  FUNCTIONS    127 


The  following  table  shows  the  changes  that  take  place  in  r  as 
6  increases  from  0  to  2  tp. 


e 

oto;. 

'!:to    '^ 
6          2 

i-T 

3         6 

se 

Oto'^ 

2 

^to^ 
2         2 

^to2. 

2.to^- 

r 

a  toO 

imag. 

Oto  a 

a  to  0 

5  TT    ,  ^   7  TT 

—  to 

6  6 

6ir  .Irr 

—  to  — 

2  2 


—    —  to  —   —  to 


e 

6         3 

i^to§^ 
3          2 

ie 

^to4. 

4.to^ 
2 

r 

0  to  a 

a  too 

IItt 
6 

11  TT 


2  2 


11  TT 


imag. 


to  2 


11 


to6  7r 


0  to  a 


The  second  column  is  to  be  read,  as  6  increases  from  0  to  ^,  3  d 

increases  from  0  to  -,  and  hence  r  decreases  from  a  to  0,  and 

similarly  for  the  other  columns. 

A  few  intermediate  values  of  r  and  6,  computed  from  a  table  of 

natural  cosines,  are  shown  for  values  of  0  ranging  from  0  to  J- 

6 


?*2  =  a"^  cos  3  8 
Fig.  103. 


128  ANALYTIC  GEOMETRY 

6        5°        10°        15°        20°        25°        30* 

r      .98  a     .93  a      .84  a      .71a      .51a        0 

Since  cos  3  6  takes  the  same  values,  either  in  the  same  order 

or  in  the  reverse  order,  when  6  increases  through  the  other 

intervals  for  which  r  is  real  as  it  does  when  0  increases  from 

0  to  — ,  the  values  of  r  are  the  same  in  those  intervals  as  in  the 
6 

iirst. 

The  curve  is  shown  in  Fig.  103.  The  dotted  portion  is  the 
part  corresponding  to  the  negative  values  of  r. 

If  6  is  allowed  to  increase  beyond  2  tt,  or  to  take  negative 
values,  cos  3  d  takes  the  same  set  of  values  over  again,  and  the 
same  points  of  the  curve  are  again  obtained. 

EXERCISE  XXVI 
Plot  the  following  curves  in  polar  coordinates. 
1.    r  =  a  sin  6.     (Prove  it  is  a  circle.)  2.  r  =  d. 

3.  r  =  a  tan  0.  4.   r  =  2  <?.  b.   r  =  a  cos  2  d, 

6.  r-  cos  ^  =  a.     (Prove  it  is  a  straight  line.) 

7.  r  sin  0  =  a.     (Prove  it  is  a  straight  line.) 

8.  r  ^  =  C.     (Called  hyperbolic  spiral.) 

9.  r—a^.      (Called  logarithmic  spiral.) 

10.  r  =  a  (1  —  cos  ^).  (The  cardioid.) 

11.  r  =  a  (1  +  cos  &).     (The  cardioid.) 

12.  r  —  f X  •     (Prove  it  is  a  parabola  by  transforming  to  rec- 

U  —  cos  Q) 

tangular  coordinates.) 

4 

13.  T  =  , = (Prove  it  is  an  ellipse. ) 

1  — .5cos0      ^ 

4 

14.  r  =- .     (Prove  it  is  an  hyperbola.) 

1  —  2  cos  ^      ^ 

15.  r  =a  sin  2  6.  16.  r  =  a  sin  Sd.  Vt.  r  =a  cos  3 0. 
18.  r  =a  cos  4 d.  19.  r"^  =  a^  cos  2  d.  20.  r^  =  a^  sin  2  0. 
21.   r2  =  a2sin3^.          22.   r^  =  a^sm(^-       23.   r  =  8  cos  ^|y 

24.    r  =asin[  — ].        25.    r  =  1  -  2  cos  5.       26.   r  =  2  -  cos  ^. 
27.    r  =  2  a  cos  ^  -f  b,  where  b  takes  the  values  0,  a,  2  a,  3  a. 


CHAPTER  IX 
PARAMETRIC   EQUATIONS   OF   LOCI 

108.  Parametric  equations.  A  single  equation  connecting 
two  variables,  which  can  be  solved  for  one  of  the  variables, 
may  always  be  replaced  by  two  equations  which  express  the 
value  of  each  of  the  variables  of  the  original  equation  in 
terms  of  a  third  variable.  Moreover,  one  of  the  two  equations 
may  have  any  form  whatever. 

Thus  in  the  equation  of  the  circle,  m?  -\-y'^=r^,  a  third  vari- 
able, t,  may  be  introduced  by  letting  x  be  equal  to  some  func- 
tion of  t ;  substituting  this  value  of  x  in  x^-\-y-  =  r^  the  value 
of  y  may  be  found  in  terms  of  t.  E.g.  ii  x  =  r  cos  t,  then 
y  =  ±  r  sin  t. 

It  often  happens  that  it  is  easier  to  obtain  the  values  of 
coordinates  of  points  on  a  given  locus  in  terms  of  some  third 
variable  than  it  is  to  obtain  an  equation  directly  connecting 
the  coordinates  of  the  points,  and  in  some  cases  the  two  equa- 
tions can  be  obtained  where  it  is  not  possible  to  obtain  the 
equation  directly  connecting  the  coordinates  of  the  points. 

The  third  variable  in  terms  of  which  the  coordinates  of  the 
points  are  expressed  is  called  the  parameter,  and  the  two 
equations  are  called  the  parametric  equations  of  the  locus. 

Frequently  the  parameter  may  be  given  an  interesting 
geometric  interpretation. 

109.  The  parametric  equations  of  the  circle.  Let  the  center 
of  the  circle  be  at  the  origin  and  let  the  radius  be  r.  Let  $  be 
the  angle  which  the  radius  to  the  point  (x,  y)  on  the  circle 
makes  with  the  ic-axis.     Then 

oc  =  r  cos  9,  y  =  r  sin  0. 

K  129 


130 


ANALYTIC  GEOMETRY 


These  equations  hold  for  every  point  on  the  circle  and  hence 
represent  the  circle  completely.  They  are  parametric  equa- 
tions of  the  circle. 

Y 


Fig.  104. 

If   0  be  eliminated  from  the  equations  (by  squaring  and 
adding),  the  ordinary  equation,  ot?  -\-y^  =  r,  is  obtained. 

110.   The  parametric  equations  of  the  ellipse.     Let  the  equa- 
tion of  ellipse  be 

^  +  ^  =  1. 

Draw  a  circle  with  center  at  origin  and  radius  a.  Through 
any  point  P{x,  y)  of  the  ellipse  draw  a  line  parallel  to  the 
?/-axis  to  meet  the  circle  in  P'  on  the  same  side  of  the  ic-axis 
as  P.  Draw  OP'  and  let  the  inclination  of  OP'  be  6.  Then 
a;  =  a  cos  6.  Substituting  a  cos  6  for  x  in  the  equation  of  the 
ellipse,  there  results  2/  =  ±  6  sin  ^.  Since  it  was  agreed  to  take 
P'  and  P  on  the  same  side  of  the  aj-axis,  the  plus  sign  must 
be  taken  in  the  value  of  y.     Hence 

a?  =  a  cos  0,  2/  =  b  sin  0, 

are  the  parametric  equations  of  the  ellipse. 
The  angle  6  is  called  the  eccentric  angle. 


PARAMETRIC  EQUATIONS  OF  LOCI  131 

Y 


Fig.  105. 

111.  Construction  of  the  ellipse.  To  construct  an  ellipse  of 
semi-axes  a  and  &,  a>b,  take  the  center  of  the  ellipse  as  a 
center  and  describe  circles  of  radii  a  and  b.  Draw  any  radius 
making  an  angle  0  with 
the  major  axis.  Through 
the  points  where  the  ra- 
dius cuts  the  inner  and 
outer  circles  draw  par- 
allels respectively  to  the 
major  and  minor  axes. 
Their  intersection  is  a 
point  of  the  ellipse. 

Proof.  Taking  the 
a;-axis  along  the  major 
axis  of  the  ellipse  the 
point  of  intersection  P 
is  at  once  seen  to  have 
the  coordinates  x  =  a  cos  6,y  =  b  sin  6,  and  is  therefore  a  point 
of  the  ellipse  from  the  preceding  article. 

Exercise  1.     Construct  an  ellipse  by  this  method. 


Fig.  106. 


132 


ANALYTIC  GEOMETRY 


Exercise  2.  Prove  that,  for  the  same  values  of  x,  the 
ordinates  of  the  ellipse  and  circle  in  Fig.  105  have  a  constant 

ratio,  -. 
a 

Exercises.     The  sun's  rays  fall  vertically  upon  a  plane; 

prove  that  the  shadow  on  this  plane  of  a  circular  hoop  not 

parallel  to  the  plane  is  an  ellipse. 

112.  The  cycloid.  The  curve  traced  by  a  fixed  point  on  the 
circumference  of  a  circle  as  the  circle  rolls  in  a  plane  along  a 
fixed  straight  line  is  called  the  cycloid. 

The  circle  is  called  the  generator  circle  and  the  point  the 
generating  point. 

To  derive  the  equations  of  the  cycloid :  Let  the  fixed  line 
be  taken  as  icaxis  and  the  point  on  this  line  where  the  gen- 
erating point  touches  it  as  the  origin.  Take  the  y-axis  per- 
pendicular to  the  ic-axis. 

Let  Pix,  y)  be  any  position  of  the  generating  point,  0  the 
angle,  measured  in  radians,  through  which  the  radius  through 
Phas  turned  since  the  generating  point  left  the  origin,  and  a 
the  radius  of  the  circle.     Then  (Fig.  107) 


0  M 


Fig.  107. 

x=OM=OH+HM. 
y  =  MP=HC+CN. 
Now  0H=  arc  HP=  aO, 

HM=  -  PN=  -  a  sin  ^, 


PARAMETRIC  EQUATIONS  OF  LOCI  133 


HC  =  a, 

CN=  —  a  cos  6. 

.'.  oc  =  a^  —a  sin  0, 
2/  =  a  —  a  cos  0. 


(1) 


(The  student  should  make  sure  that  these  equations  hold 
for  either  position  of  the  generator  circle  shown  in  Fig.  107, 
and  should  draw  other  positions  of  the  generator  circle  and 
prove  that  the  same  equations  hold.) 

Equations  (1)  give  the  values  of  x  and  y  in  terms  of  a  third 
variable  Q.  By  assigning  values  to  ^,  values  of  x  and*?/  may 
be  computed  and  thus  points  on  the  curve  located. 

It  is  usual  to  take  the  two  equations  (1)  as  representing  the 
cycloid,  but  a  single  equation  connecting  x  and  y  may  be  ob- 
tained as  follows : 

From  the  second  equation,  1  —  cos  ^  =  ^ ,  or  vers  ^  =  ^  • 

a  a 

.-.  ^=:vers~^^, 
a 

and         sin  B  =  Vl-cos2^=  ^1  -  f^^~^X  =  ^  V2ay-y\ 

Substituting  these  values  of  6  and  sin  $  in  the  first  of  eqs. 
(1),  there  results 

x  =  a  yers"^  -  if  V2  ay  —  y\  (2) 

113.  Construction  of  the  cycloid.  Besides  the  method  of 
locating  points  on  the  cycloid  by  computing  values  of  x  and  y 
from  eqs.  (1)  of  Art.  112,  the  following  method  may  be  easily 
employed :  On  a  straight  line  lay  off  a  distance  OA  equal  to 
the  circumference  of  the  generating  circle.  At  the  middle 
point  B  of  OA  draw  a  circle  equal  to  the  generating  circle 
tangent  to  OA.  Divide  OB  into  a  number  of  equal  parts  by 
the  points  Bi,  B,,,  -Bg,  etc.,  and  the  semi-circumference  BC  into 
the  same  number  of  equal  parts  by  the  points  (7i,  Og,  Cg,  etc., 


134 


ANALYTIC  GEOMETRY 


obtained  by  use   of  the   protractor.     Through    Ci,   Cg,  Cg, 
draw  lines  parallel  to  OA. 

-Pi-        C 


C^  0  ■-" 


O  B2  B,   B 

Fig.  108. 

As  the  circle  rolls  back,  the  point  P,  now  at  the  top  of  the  cir- 
cle, generates  the  cycloid,  the  point  P  descending  to  the  level 
of  Ci  when  the  point  of  tan  gen  cy  moves  back  to  Bi.  Hence 
the  point  Pj  may  be  obtained  by  using  Ci  as  a  center  and  BBi 
as  a  radius  to  describe  an  arc  cutting  the  line  through  Ci. 

Similarly  with  radius  equal  to  BBo  and  center  C2  the  point 
P2  is  obtained,  etc. 

Other  methods  of  constructing  the  cycloLl  are  employed  by 
draftsmen. 

Exercise  1.  Construct  a  cycloid  by  the  method  explained, 
dividing  the  circumference  into  twelve  equal  parts. 

Exercise  2.  Construct  a  cycloid  by  computing  values  of 
X  and  y  by  eqs.  (1),  Art.  112. 

114.  The  hypocycloid.  The  hypocycloid  is  the  curve  traced 
by  a  fixed  point  on  the  circumference  of  a  circle  which  rolls 
internally  along  the  circumference  of  a  fixed  circle. 

To  derive  the  equations  of  the  hypocycloid :  Let  the  radii 
of  the  fixed  and  rolling  circles  be  a  and  b  respectively.  Take 
the  center  of  the  fixed  circle  as  origin,  and  the  line  through 
this  center  and  the  point  of  contact  of  the  generating  point 
with  the  fixed  circle  as  a;-axis.  Let  P(x,  y)  be  any  position  of 
the  generating  point,  d  the  angle  through  which  the  line 
of  centers  has  rotated,  and  <^  the  angle  through  which  any 


PARAMETRIC  EQUATIONS  OF  LOCI 


135 


radius  of  the  generator  circle  has  turned  since  the  generating 
point  left  the  x-axis.     Then  (Fig.  109), 

X  =  0M=  OH -{-NP=OC  cos  e+CF  cos  <l> 

=  (a  —  b)  cos6-\-b  cos  <f>, 

y  =  MP=  HC-NC={a-  b)  sinO-b  sin  <^. 


Fig.  109. 


Now  arc  PB  =  arc  AB,  and  therefore  b(<f>-{-0)  =  aO, 

,      a  —  b 
9  =  - 


or 


0. 


.'.  0?=  (a  — 6)cos0  +  b  cosi— -0); 


115.  Construction  of  the  hypocycloid.  From  the  above 
equations  as  many  values  of  x  and  y  as  desired  may  be  com- 
puted by  assigning  arbitrary  values  to  6.  By  this  means  a 
sufficient  number  of  points  may  be  obtained,  through  which 
the  curve  may  be  drawn. 

Another  method  is  as  follows :  Draw  two  concentric  circles, 
K  and  K',  with  radii  a  and  a  —  b  respectively.     Let  <^'=  ^  -|-  <^ ; 


136 


ANALYTIC  GEOMETRY 


then  ad=  b<l>'.     Compute  the  value  of  6  which  makes  cf>'  =  360° 

i.e.  0  =  -  360°.     Let  AOB  be  this  angle,  constructed  by  use  of 
a 

the  protractor.     Then  B  is  the  second  point  of  contact  of  the 

generating  point  with  the  fixed  circle.     Divide  AOB  into  any 

number,  n,  of  equal  parts  and  draw  radii  to  intersect  the  circle 

K'  at  Ci,  Co,  Cs,  etc.,  and  the  circle  K  at  Bi,  B2,  B^,  etc.     With 

Ci,  C2,  •"  as  centers  draw  circles  of  radius  b. 


Fig.  110. 

The  position  of  the  generating  point  on  the  first  of  these 

circles  is  obtained  by  drawing  an  angle  B^C^P^  equal  to  -th  of 

n 

360°;  the  point  on  the  second  circle  by  drawing  an  angle 
B^C^P^  equal  to  ?ths  of  360°,  etc.     See  Fig.  110,  where  n  =  8. 

116.   The  hypocycloid  where  a  =  2  6.     Letting  a  =  2  6  in  the 
equations  of  Art.  114,  there  is  obtained 

a?  =  a  cos  0, 

The  latter  equation  shows  that  the  generating  point  moves 
along  the  a>-axis,  and  the  former  that  it  is  at  any  time  in  the 
same  vertical  as  the  point  of  contact  of  the  circles. 


PARAMETRIC  EQUATIONS  OF  LOCI  137 

Hence,  if  a  circle  rolls  within  a  Jixed  circle  of  double  the  diam- 
eter,  every  point  of  the  rolling  circle  moves  hack  and  forth  along 
a  diameter  of  the  fixed  circle.  Moreover,  if  the  circle  rolls  with 
uniform  angular  velocity,  every  point  of  it  moves  with  simple 
harmonic  motion.* 

117.  The  four-cusped  hypocycloid.  The  points  where  the 
generating  point  reverses  its  direction  of  motion  are  called 
cusps.  Thus  the  points  of  contact  of  the  generating  point  and 
the  fixed  circle  are  cusps. 

If  a  =  4  5  there  are  four  cusps.  The  curve  in  this  case  is  of 
interest  because  it  is  possible  to  eliminate  0  between  the  equa- 
tions of  Art.  114  and  obtain  a  simple  equation  connecting  x 
and  y. 

Substituting  -  for  b  in  eqs.  (1),  Art.  114,  they  become 

a;  =  — cos^  +  |cos3d  =  |(3cos(9-f  cos3^), 

2/  =  —  sin  ^-  -  sin  3  (9  =  -(3  sin  0  -  sin3  ^). 
4  4  4 

By  trigonometry, 

cos  3  ^  =  4  cos^  ^  —  3  cos  0, 
sin3(9  =  3sin^-4sin3d. 
Substituting  these  values,  there  result 
jc=acos^^, 
y  =  asm^O, 


from  which 


cos^=f-]  > 
a, 


sm6/  = 


*  When  a  point  moves  with  uniform  velocity  along  the  circumference 
of  a  circle  the  projection  of  the  point  on  any  diameter  is  said  to  have 
simple  harmonic  motion. 


138  ANALYTIC  GEOMETRY 

Squaring,  adding,  and  clearing  of  fractions, 

118.  The  epicycloid.  The  epicycloid  is  the  curve  traced  by 
a  fixed  point  on  a  circle  which  rolls  externally  on  the  circum- 
ference of  a  fixed  circle. 


Fig.  111. 


Let  the  student  show  from  the  figure  that  the  equations  are 
ac  =  (a  +  6)cos  6  —ft  cos 


2/  =  (a  +  6)siii  e  -  6  sin 


b 
a-\-b, 


Notice  that  the  equations  differ 
from  those  of  the  hypocycloid  only 
in  having  —  b  take  the  place  of  b. 

119.  The  cardioid.  The  epicy- 
cloid for  which  the  rolling  and  fixed 
circles  are  equal  is  called  the  car- 
dioid. Its  equations  are  obtained  by 
letting  6  =  a  in  the  equations  of  the 


Fig.  112. 


PARAMETRIC  EQUATIONS  OF  LOCI 


139 


Fig.  113. 


preceding  article.     They  then  become 

a?  =  2  a  cos  0  —  a  cos  2  9, 
2/  =  2  a  sin  0  —  a  sin  2  e. 

120.  The  involute  of  the  circle.  If  a  thread  is  wound  around 
a  circular  form  and  then  unwound,  kept  always  stretched,  any 
point  of  the  thread  traces  a 
curve  called  the  involute  of  the 
circle. 

To  derive  its  equations: 
Choose  the  axes  as  in  Fig.  113. 
Let  a  be  the  radius  of  the 
circle,  P(x,  y)  the  position  of 
the  generating  point  at  any 
time,  and  6  the  angle  through 
which  the  radius  to  the  point 
of  tangency  has  turned  during 
the  unwinding.     Then 

x=  0M=  0N+  NM=  0N+  TP  sin  0, 

y  =  MP  =  NT-  JST=  NT-  TPcos  6. 

Now  TP  =  arc  ^  r  =  aO. 

.*.  a?  =  a  cos  0  +  a  0  sin  0, 
2/  =  a  sin  e  —  a  0  cos  0, 

are  the  equations  of  the  involute  of  the  circle. 

EXERCISE   XXVII 

1.  Trove  that  if  a  circle  of  radius  a  rolls  along  a  straight  line,  a  point 
on  a  fixed  radius  of  the  circle  at  a  distance  h  from  the  center  describes  a 
curve  whose  equations  are  * 

jc  =  a^  —  &  sin  ^,  y  =  a  —  hco&d. 
Plot  the  curve  for  6  <  a  ;  for  ?)  >  a. 
These  curves  are  called  trochoids. 

2.  Devise  a  method  of  constructing  the  cycloid  similar  to  the  method 
of  constructing  the  hypocycloid  in  Art.  115. 


140  ANALYTIC  GEOMETRY 

3.  Carefully  construct  on  coordinate  paper  a  cycloid  by  the  method 
you  have  described. 

By  counting  the  squares  between  the  cycloid  and  the  line  on  which  the 

circle  rolls  and  the  squares  in  the  generating  circle,  what  idea  do  you  get 
of  the  area  of  the  cycloid  ? 

4.  By  combining  the  equations  of  the  cardioid  (Art.  119)  and  trans- 
forming to  polar  coordinates,  show  that  the  polar  equation  of  the  cardioid 
is  r  =  2  aCl  —  cos  0),  where  the  pole  is  the  point  of  contact  of  the  gen- 
erating point  with  the  fixed  circle. 


Fig.  114. 

Suggestion.  Square  and  add  the  equations  of  Art.  119,  move  to  new 
origin  by  letting  x  =  x'  +  a,  y  =  y' ;  substitute  x'  =  r  cos^,  y'  =  rsin  6 ; 
complete  the  square  in  the  terms  in  r,  and  extract  the  square  root.  Also 
derive  the  polar  equation  independently  from  the  figure.     (Fig.  114.) 

5.  Taking  the  origin  at  the  point  of  the  cycloid  farthest  from  the 
line  on  which  the  circle  rolls,  and  the  ic-axis  parallel  to  that  line  show  that 
the  equations  of  the  cycloid  are 

X  =  ad  +  a  sin  0,  y  z=—  a  -\-  a  cos  6, 

where  6  is  measured  from  the  positive  direction  of  the  y-axis  to  the  radius 
of  the  circle  through  (a;,  y),  clockwise  rotation  being  counted  positive. 

6.  Construct  a  hypocycloid  where  a  =  Sb. 

7.  Devise  a  method  for  constructing  the  epicycloid  and  apply  it  to 
the  case  where  a  =  4b. 


PARAMETRIC  EQUATIONS  OF  LOCI 


141 


8.  Construct  the  involute  of  a  circle. 

9.  A  circle  rolls  along  a  straight  line,  and  a  line  through  the  center  of 
the  circle  turns  about  a  point  of  the  fixed  line.  Find  the  equations  of  the 
locus  of  the  point  of  intersection  of  line  and  circle,  and  plot  the  curve. 

Ans.   x  =  a  (cot  6  +  cos  ff) , 

y  =  a{\  +  sin  ^)  for  outer  point, 

X  =  a(c,otd  —  cos^), 

y  =  a(\  —simd)  for  the  inner  point. 


and 


Fig.  115. 

10.  Show  that  the  polar  equations  of  the  curves  of  example  9  are 
r  =  a  (esc  ^4-1)  and  r  =  a  (esc  ^  —  1)  respectively. 

11.  A  circle  moves  with  its  center  always  on  a  straight  line,  and  a 

second  straight  line  passes 
through  the  center  of  the  circle 
and  a  fixed  point.  Find  the 
loci  of  the  pointp  of  intersection 
of  the  second  line  and  the  circle. 

Ans.   Using  the  notation  of 
Fig.  116, 

X  =  b tan  0  -{■  asm 6, 

y  =  acos 6,  for  P. 

X  =  b  tan  d  —  a  sin  0, 

Fig.  116.  y  =—  acos0,  for  P'. 

12.  Plot  the  curves  of  example  11  for  6  <  a,  6  =  a,  &  >  a. 


CHAPTER   X 


INTERSECTIONS  OF  CURVES.    SLOPE  EQUATIONS  OF 
TANGENTS 

121.  Intersections  of  curves.  It  has  been  seen  that  an 
equation  in  two  variables  can  be  represented  graphically  by  a 
curve,  every  point  of  which  has  coordinates  which  satisfy  the 
equation.  Two  different  equations  in  the  same  two  variables 
will  then  in  general  represent  two  different  curves.  If  these 
curves  be  plotted  on  the  same  diagram  they  may  or  may  not 
intersect.  The  coordinates  of  the  points  of  intersection,  if  any, 
must  satisfy  both  equations,  and  no  other  points  will  have 
this  property.  ISTow  the  values  of  the  variables  which  satisfy 
two  equations  are  obtained  by  solving  the  two  equations  as 
simultaneous.     Hence  to  find  the  points  of  intersection  of  two 

curves,  solve  the  equa- 
Y  tions   of   the  curves  as 

simultaneous.  The  real 
values  of  the  variables 
so  obtained  which  sat- 
isfy both  equations  are 
the  coordinates  of  the 
points  of  intersection  of 
the  curves. 

Example.  To  find 
the  points  of  inter- 
section of  the  circle 
ic2  -f  2/2  =  16  and  the  pa- 
rabola x^  =  Qy.  Elimi- 
nating X  from  the  first 
Fig.  117.  equation    by    substitut- 

142 


INTERSECTIONS  OF  CURVES  143 

ing  the  value  of  x  from  the  second,  there  results 

from  which  y  =  2  or  —  8.  Substitutmg  the  first  of  these  values 
of  y  in  the  second  equation,  there  is  obtained  x=  ±  2 V3. 
The  substitution  of  —  8  in  the  second  equation  gives  imaginary 
values  of  x.  Hence  the  points  of  intersection  are  (2v^,  2) 
and  (-2V3,  2),  or  approximately   (3.46,  2)  and  (-  3.46,  2). 

On  plotting  the  curves  these  results  are  seen  to  be  approxi- 
mately correct. 

EXERCISE  XXVm 

Find  the  points  of  intersection  of  the  following  pairs  of  curves.  Check 
graphically  by  plotting  the  curves  and  measuring  the  coordinates  of  the 
points  of  intersection. 

1.  y?  +  yp-  =  5,  y^  =i\x. 

2.  y  =  3  ic  +  7,  x2  +  ?/2  :=  9. 

3.  (a)  y  =  2  a;  +  i,      y2  =  4  a;.  Ans.  {\,  1)  (Tangent) . 

(6)  y  =  2a;  +  .49,  y'^  =  ^x.  Ans.  (.326,  1.141),  (.184,  .859). 

(c)  y  =  2a;  4-  -51,  y^  =  \x.  Ans.  No  intersection. 

4.  a;2  +  4  ?/2  -  16,  a;2  +  y  =  0. 

5.  3  a;  -  y  =  1,  16  a;2  +  9  y^  =  144. 

6.  a; +2/ =  5,  9  a;2  +  16  2/2  =  144. 

7.  a;2  -I-  ?/2  =  16,  a:2  -  2/2  =  9. 

8.  For  what  values  of6isz/  =  2a;  +  6  tangent  to  a;2  +  y2  =;  9  ? 

9.  For  what  values  of  6  is  y  =  wa;  +  6  tangent  to    a;2  -f-  y2  _  ,.2  9 

10.  For  what  value  of  p  is  y2  =  2  px  tangent  to?/  =  3a;4-l? 

11.  Prove  that  the  two  segments  of  any  line  which  cuts  xy  —  O  in  two 
points,  included  between  the  curve  and  its  asymptotes,  are  equal. 

122.  Graphical  solution  of  simultaneous  equations.  It  fre- 
quently happens  that  when  two  equations  containing  two 
variables  are  given  it  is  not  possible  to  eliminate  one  of  the 
variables,  and  so  obtain  an  equation  with  only  one  variable  ; 
or,  if  the  elimination  is  possible,  the  resulting  equation  may 
be  very  difficult  or  impossible  of  solution  by  ordinary  methods. 


144 


ANALYTIC  GEOMETRY 


In  such  cases,  if  the  coefficients  are  numerical,  an  approximate 
solution  may  be  obtained  by  carefully  plotting  the  curves  and 
measuring  the  coordinates  of  the  points  of  intersection.  More 
accurate  solutions  may  then  be  obtained  by  methods  illustrated 
in  the  following  examples. 

Example  1.     To  find  the  intersection  of  the  curves 


and 


y  =  sin  X 
y  =  2x  +  l. 


(1) 
(2) 


Plot  the  curves  carefully  on  coordinate  paper. 

From  the  figure  the  abscissa  of  the  point  of  intersection  is 
seen  to  be  about  —  .9.  Substitute  this  value  in  equations  (1) 
and  (2),  remembering  that  .9  radian  =  .9  of  57°.3  =  51°34',  and 
there  results, 

from  (1)  y  =  sin(-  5r34')  =  -  .78, 

from  (2)  2/  =  -  -8. 

Y 


Fig.  118. 

This  shows  the  assumed  value  of  x  to  be  too  small,  but  very 
near  to  the  correct  value.     (Compare  Fig.  118.) 
Try  next  a  =  -  .88. 

Then,  from  (1),        y  =  sin(-  50°25')  =  -  .771, 

from  (2),         2/  =  -  •'J'^. 
This  shows  the  assumed  value  of  x  to  be  too  large,  so  n^xt  try 
x=-  .89. 


INTERSECTIONS  OF  CURVES  145 

Then,  from  (1),         y  =  sin(-  51°)=  -.777, 
from  (2),         y=-  .78. 

Hence,  correct  to  two  significant  figures,  the  solution  is 

x=-  .89,     y=-  .78. 

Example  2.     To  solve  the  equation 

a^_2a;2  +  4a;-7=0.  (1) 

Let  ?/  =  a^-2a^  +  4a;-7.  '       (2) 

Then  the  solution  of  (1)  is  the  same  as  the  simultaneous  solu- 
tions of  (2)  and  the  equation 

2^  =  0.  (3) 

Plot  the  curve  of  eq.  (2).     (Figure  not  shown.) 
The  following  are  corresponding  values  of  x  and  y: 


X 

0 

1 

2 

3 

4 

-1 

-2 

-3 

y 

-7 

-4 

1 

14 

41 

-14 

-31 

-64 

The  curve,  is  seen  to  cross  the  a^axis  between  1  and  2,  at 
about  1.8. 

Try  this  value  of  x  in  eq.  (2) ; 

y  =  5.832  -  6.48  +  7.2  -  7  =  -  .448. 

Hence  the  value  of  1.8  for  x  is  too  small. 

Try  next  a;  =  1.9 ;  then  y  =  .239. 

Hence  the  value  of  1.9  for  x  is  too  large. 

Plot  now  on  an  enlarged  scale  the  points  representing  x  and 
2/  for  a;  =  1.8  and  1.9,  and  join  the  points  by  a  straight  line. 
Since  the  interval  is  small,  the  curve  probably  differs  but 
slightly  from  a  straight  line  in  the  interval.  The  line  is  seen 
to  cross  at  about  .65  of  the  distance  from  1.8  to  1.9.  Then 
1.865  is  probably  a  close  approximation  to  a  root  of  eq.  (1). 
Substituting  this  value  of  x  in  eq.  (2),  there  results  ?/=  — .0094. 


146  ANALYTIC  GEOMETRY 

The   work  of   computation,   arranged   according   to   Horner's 
method  of  synthetic  division,  is  as  follows : 
1-2                    4-7  )1.865 
■    1.865           -.2518 6.9906 

-  .135         -3.7482  -    .0094 

By  the  Remainder  Theorem  from  Algebra,  the  value  of  2/  is 
the  last  remainder,  —  .0094. 

Since  y  comes  out  negative,  it  shows  that  in  this  case  the 
assumed  value  of  x  is  too  small.     Try  then  x  =  1.866. 
1-2                    4-7  )1.866 
1.866           -  .2500 6.9975 

-  .134  3.7500  -    .0025 
Hence  y=-  .0025. 

Try  next  x  =  1.867 : 

1-2  4-7                     )1.867 

1.867  -.2483 7.0044 

.133  3.7517                  .0044 

Hence  y  =  .0044. 

The  root  therefore  lies  between  1.866  and  1.867  and  is  nearer 
to  the  former.  Hence,  correct  to  four  significant  figures,  a  root 
of  eq.  (1)  is  1.866. 

Evidently  one  could  by  this  method  obtain  a  root  correct  to 
any  desired  degree  of  accuracy. 

Example  3.   To  solve  the  equation 

<^2  _  sin  2  <^  =  0.  (1) 

This  may  be  treated  as  in  the  last  example,  or  it  may  be 
more  easily  solved  as  follows :  Plot  separately  the  curves 

y  =  ^'  (2) 

and  2/  =  sin  2  <^  (3) 

on  the  same  diagram.     Then  a  value  of  <^  at  a  point  of  inter- 
section of  the  curves  of  eqs.  (2)  and  (3)  is  a  root  of  eq.  (1). 


INTERSECTIONS  OF  CURVES 


147 


The  figure  shows  that  a  value  of  <^  at  the  intersection  is  a 
little  less  than  1.     Try  then  <^  =  .9. 

Then  from  (2)  y=     M 
and  from         (3)  y  =     .974 
Difference  =—.164. 

Y 


Fia.  119. 


Substitute  <f>  =  ^f 
then  from  (2)  y  =  1 
and  from        (3)  y  =    .909 

Difference  =    .091 


Plot  on  an  enlarged  scale  the  difference  for  <^  =  .9  and  <^  =  1, 
using  <^  as  abscissa  and  difference  as  ordinate,  and  connect  the 
points  obtained  by  a  straight  line.  This  straight  line  is  seen 
to  cross  the  axis  at  about  .65  of  the  distance  from  .9  to  1.  On 
substituting  <^  =  .965  there  results 

from         (2)  y  =  .931 
and  from         (3)  y  =  .936 
Difference  =—  .005 

Let  the  student  show  that  when  <f)  =  .966  and  .967  the  differ- 
ences computed  as  above  are  —  .0022  and  .0004,  respectively, 


148 


ANALYTIC  GEOMETRY 


and  that  hence  the  solution  of  eq.  (1),  correct  to  three  significant 
figures,  is  <^  =  .967. 


EXERCISE  XXIX 


Solve  the  following  pairs  of  equations  : 


cos  X,   1/2  =  4:X. 


2.   10y  =  x,   y  =  \ogiox. 

4.  X  =  ^  —  sin  ^,   x  =  l  —  cos 


3.   s  ==  sin  3  ?,    s  =  tan  2 1. 

5.    y  =  x^,  y  =  2^. 

Solve  the  following  equations  by  graphical  methods  : 

6.    x8  +  5  =  0.  7.    a;8  _  X  +  7  =  0. 

8.    2  61  -  cos  2  (9  ^  0.  9.    1  -  $  -  tSiU  d  =  0. 

10.   2=^  -  X  +  1  =  0. 

123.   Slope  equations  of  tangents.     Tangent  to  the  ellipse. 

Let  a  line  of  slope  7/1  be  drawn  tangent  to  the  ellipse 


(1) 


^  +  ^  =  L 

To  derive  its  equation. 

Any  line  of  slope  m  has  an  equation  of  the  form 

y  =  mx  4-  k.  (2) 

If  eqs.  (1)  and  (2)  be  solved  as  simultaneous,  the  points  of  in- 
tersection of  the  loci 
will  be  obtained. 
These  intersections 
may  be  real  and  dis- 
tinct, real  and  co- 
incident, or  imagin- 
ary, depending  upon 
the  value  of  7c.  It 
is  evident  from  the 
figure  that  there  are 
two  values  of  k  for 
which  the  line  is  tan- 
gent to  the  ellipse. 


Fig.  120. 


SLOPE  EQUATIONS  OF  TANGENTS  149 

Substituting  the  value  of  y  from  eq.  (2)  in  eq.  (1)  and  collecting 
terms,  there  results 

(52  4-  a?m')x'  +  2  a'mkx  +  aXlc"  -b')=0.  (3) 

The  roots  of  eq,  (3)  are  the  abscissas  of  the  points  of  intersec- 
tion of  the  line  and  the  ellipse.  In  order  that  the  line  be 
tangent  to  the  ellipse  these  values  of  x  must  be  equal ;  and 
conversely,  if  they  are  equal,  so  also  are  the  values  of  y  obtained 
by  substituting  these  values  of  x  in  eq.  (2),  and  hence  the  line 
is  a  tangent.  Now  the  condition  that  the  roots  of  the  equation 
ax^  -\-bx-{-c  =  0  be  equal  is  6^  =  4  ac.  Hence,  the  roots  of 
eq.  (3)  are  equal  if 

4  aWli^  =  4  a^  (A;2  _  52^^  q,2  _^  ^2^2^^ 

which  reduces  to 

Therefore  the  equations  of  the  tangents  to  the  ellipse 


s^.y^ 


with  slope  m  are 


y  =  mac  ±  Va^m^  +  b^. 


These  equations  are  called  the  slope  equations  of  the  tangents 
to  the  ellipse. 

124.   Tangent  equations  for  reference.     The  student  should 
derive  the  following  equations  of  tangents  to  the  given  curves. 

Tangent 


Curve 

(1) 

a?2  + 

V^  = 

:r2, 

(2) 

^ 

a^ 

62 

1, 

(3) 

a2 

62 

1, 

(4) 

y2 
62 

-1, 

y  =  tnx  ±  rwrrtP'  +  1. 


y  =  tnsc  ±  Va^n^  +  62. 
y  =  mx  ±  Va'hn^  —  62. 


y  =  mx  ±  V62  —  a'^m'^ 


150  ANALYTIC  GEOMETRY 

(5)    2/2  =  2j>x,  2,  =  m^  +  -^ 


(6)    oo^^^py,  y  =  fna^-:i^l^. 


2fn 


2 

(7)xy  =  c,  y  =  mx±  ^yZ—cm. 

EXERCISE  XXX 

(Use  the  above  formulas  in  solving  these  exercises.) 

1.  Find  the  equations  of  tangents  to  ic^  -\-y^  =  \Q  which  have  a  slope 
equal  to  V3.     Check  graphically. 

2.  Find  the  equations  of  tangents  to  9  a;2  +  16  y"^  =  576  which  are 
parallel  toy  =  x.    Check  graphically. 

3.  Find  the  equation  of  a  tangent  to  ?/2  =  6  x  which  is  perpendicular 
to2x— y  —  3  =  0.    Plot  the  lines.     Where  do  they  intersect  ? 

4.  Write  the  equation  of  a  tangent  to  y-  =  2px  and  the  equation  of  a 
line  through  the  focus  perpendicular  to  the  tangent,  and  prove  that  they 
intersect  on  the  ?/-axis. 

5.  Obtain  the  slope  equation  of  a  tangent  to  the  circle  from  the  equa- 
tion of  the  tangent  to  the  ellipse. 

6.  Find  the  equations  of  tangents  to  y^  —  Qx  from  the  exterior  point 
(2,  4).     Check  graphically. 

7.  Find  the  equations  of  tangents  from  (7,  1)  to  x^  +  2/2  _  25.  Check 
graphically.  Ans.  3a;  +  4y  —  25  =  0,  4a;  —  3y  —  25  =  0. 

8.  Find  the  equations  of  tangents  to  9  x^  _  25  y'^  =  225  which  pass 
through  (—  1,  3).     Check  graphically. 

Ans.   X  — ?/+4  =  0,  3x  +  42/— 9  =  0. 

9.  Find  the  equations  of  tangents  to  12  x^  +  5  ?/2  =  30  which  intersect 
in  (—3,  —2).     Check  graphically. 

10.  Show  by  the  use  of  formula  (7),  Art.  124,  that  no  tangent  can  be 
drawn  to  xy  =  8  which  has  a  positive  slope. 

11.  Find  the  equations  of  all  lines  that  are  tangent  to  x^  +  y2  =  25  and 
x2  +  4?/2  =  36.     Plot.  Ans.    11  y  =±4\/ll  x  ±  15V33. 

12.  Find  the  equation  of  a  common  tangent  to  y"^  =  2px  and  x^  =  2py. 
Check  graphically. 

13.  Find  the  equation  of  the  common  tangent  to  y^  =  6  x  and  x^  =  48  y. 
Check  graphically. 


SLOPE  EQUATIONS  OF  TANGENTS  151 

14.  Find  the  equations  of  tangents  to  h^x"^  —  a^y^  =  a%^  that  intersect 
in  the  origin.  Ans.  The  asymptotes,  hx  —  ay  —  0,  hx  +  ay  =  0. 

15.  For  what  value  of  m\^y  =  mx  +  8  tangent  to  y"^  =  ix^    Plot. 

16.  A  line  is  tangent  to  x^  -\-y'^  =  16  and  y'^  =  Qx;  find  its  equation. 
How  many  solutions  ?    Plot. 

17.  Find  the  equations  of  lines  of  slope  2  which  are  tangent  to 

a:2  4.y2_4^_l_6y_l_5,3  0.    Plot. 

18.  Prove  that  y  —  k  =  m^x  —  h)  ±  r\'l  -\-  m-  '\^  tangent  to 

Suggestion.     Move  the  origin  to  (h,  k)  ;  use  formula  (1),  Art.  124,  and 
then  translate  the  axes  to  the  original  position. 

19.  Find  the  equations  of  tangents  to  x^  +  ?/2  —  4  x  +  6 1/  —  12  =  0,  with 
slope  2,  by  using  the  formula  of  Ex.  18. 

20.  Prove  that  y  —  k  =  m{x  —  h)  +  -^  is  tangent  to 

2m 

(y-ky  =  2p{x-h)' 

21.  Find  the  equation  of  a  tangent  to  ?/2  —  2?/  —  4x  =  0  with  a  slope 
equal  to  3. 

22.  Find  thp  slope  equation  of  a  tangent  to  ^^  ~  ^^^   +  ^^  ~  ^')   =  1. 

a^  62 

23.  Find  the  equations  of  tangents  to  4x2  +  9y2-|-8x-36i/  +  4  =  0 
with  slope  equal  to  —  3. 

24.  Find  the  equations  of  lines  with  slope  equal  to  2  which  are  tangent 
to  aj2  -  y2  =  1. 

25.  Prove  that  a  line  with  slope  numerically  less  than  -  cannot  be 

a 

tangent  to  62^2  -  a'^y'^  =  a%\ 

26.  Prove  that  any  two  tangents  to  y^  =  2px  which  are  at  right  angles 

to  each  other  intersect  on  the  line  x  =  —^,  the  directrix. 

2' 

27.  Show  that  any  two  tangents  to  the  ellipse  h'^x'^  +  ci^y^  =  a%^  which 
are  perpendicular  to  each  other  intersect  on  the  circle  ^2  -\-y'^  =  «2  -\.  ^2^ 

Suggestion.     The  equations  of  two  tangents  to  the  ellipse  which  are 
perpendicular  to  each  other  are 

y  =  mx-{-  y/m'^a^  +  6^,  (1) 

and  y  =  -^-  +  A/-,+  &'-  (2) 


152 


ANALYTIC  GEOMETRY 


If  these  equations  be  regarded  as  simultaneous,  the  values  of  x  and  g 
that  satisfy  thera  are  the  coordinates  of  the  intersection  of  tlie  two  tangents. 
If  then  m  be  eliminated  between  the  two  equations,  an  equation  will  be 
obtained  which  is  satisfied  by  the  coordinates  of  the  intersection  of  any 
two  perpendicular  tangents. 

In  this  case  the  elimination  is  easily  made  as  follows  ;  Eqs.  (1)  and  (2) 
may  be  written 

y  —  mx—  y/m'^cfi  +  h^t 
and  my  -\-x=  y/d^  +  rri^h^. 

Then  square  and  add, 

28.  Prove  that  the  locus  of  foot  of  the  perpendicular  from  the  focus  upon 
a  tangent  to  the  ellipse  V^x'^  +  a^y'^  =  d^lP-  is  the  circle  aj^  +  y2  _  ^^2^  Check 
graphically. 

29.  Show  that  any  two  tangents  to  the  hyperbola  hH^  -  a?-y'^  =  a^W' 
which  are  perpendicular  to  each  other  intersect  upon  the  circle 
a;2  _|_  ^2  _  Q,2  _  yi^  if  a>h,  but  that  there  are  no  perpendicular  tangents 
if  a  <  &.    What  if  a  =  &  ? 

30.  Prove  that  the  locus  of  the  foot  of  the  perpendicular  from  the  focus 
upon  a  tangent  to  V^x'^  —  aV  =  «^&"^  is  the  circle  x^  +  2/2  =  (fi^  Check 
graphically. 

31.  Find  the  equation  of  the  locus  of  the  foot  of  the  perpendicular  from 
the  center  upon  the  tangent  to  W-x^  +  a^y^  —  a^W-. 

Ans.     (x2  +  !/2)2  =  a2x2  +  62^,2, 

32.  By  transforming  to  polar  coordinates  reduce  the  equation  of  Ex.  31 
to  the  form  r^  =  oP-  cos2  0  +  IT-  sin2  Q. 

Construct  the  curve  by  use 
of  the  circles  r  =  a  cos  6^  and 
r=.hsva.d.  (See  Fig.  121.) 
33.  Find  the  equation  of 
the  locus  of  the  foot  of  the 
perpendicular  from  the  cen- 
ter upon  a  tangent  to  the 
equilateral  hyperbola  x'^  —  y^ 
=  a?    Ans. 

(x^  +  y-^y  =  a^Cx^-y^). 
Yjq  121  ^^-    Show  that  the  equa- 

tion of  the  locus  of  Ex.  33  in 
polar  coordinates  is  r^  =  a^  cos  2  d.  Plot  the  curve.  This  cui-ve  is  called 
the  lemniscate. 


CHAPTER   XI 


SLOPES.     TANGENTS   AND    NORMALS.     DERIVATIVES 

125.  Introduction.  In  this  chapter  methods  will  be  derived 
of  finding  the  direction  of  a  curve  whose  equation  is  known  in 
rectangular  coordinates  at  any  point  of  the  curve ;  of  finding 
the  equations  of  tangent  and  normal  to  the  curve  at  any  point ; 
and  some  general  methods  established  which  will  shorten  the 
work  of  computing  the  slopes  of  curves.  These  methods  will 
be  shown  in  their  application  to  some  numerical  cases. 

126.  Increments.     In  an  equation  connecting  x  and  ?/,  e.g. 

42/  =  aj2-2a;  +  4,  (1) 

if  a  value  be  assigned  to  x,  y  takes  a  value  to  correspond ;  and 
if  X  is  given  a  different  value,  y  will  in  general  take  a  different 
value. 

Thus,  if  x  =  0,  then  2/  =  1 ;  if  a?  =  1,  then  y  =  %\  if  x  =  —  1, 
then  2/  =  J ;  if  'ic  =  2,  then  y  =  l. 

Any  change  in  x  in  general  brings  about  a  change  in  y. 

These  changes  are  most  easily  seen  by  referring  to  the  curve 
which  eq.  (1)  represents. 
As  the  point  {x,  y)  traces 
the  curve,  both  x  and  y 
change,  and  the  amount 
that  y  changes  depends 
upon  the  amount  that  x 
changes,  and  also  upon 
the  point  of  the  curve  from 
which  the  change  is  reck- 
oned. Thus,  if  X  increases 
by  1  from  the  value  1,  y 
increases  from  J  to  1,  or 

163 


154 


ANALYTIC  GEOMETRY 


the  increase  in  ?/  is  J ;  while  if  x  increases  by  1  from  the  value 
2,  y  increases  from  1  to  J,  or  the  increase  in  y  is  f .  Again,  if  x 
increases  from  —  3  to  —2,y  decreases  from  -L^-  to  -^-j  or  it  may- 
be said  that  the  increase  in  y\^—\. 

Suppose  now  that  some  definite  value  of  x  is  chosen,  and  a 
study  made  of  the  changes  brought  about  in  y  by  increasing 
X  by  small  amounts  from  this  definite  value.  Let  the  increase 
that  is  given  to  x  be  denoted  by  the  symbol  Aa?,  read  "  delta  oc^ 
or  "  increment  oo  " ;  and  let  the  increase  brought  about  in  y  by 
this  change  in  ic  be  denoted  by  Ay,  read  "delta  2/,"  or  "incre- 
ment yP 

The  following  table  shows  values  of  a;,  ?/,  Aa;,  Ai/,  and  the 

ratio  — ^,  the  value  2  being  chosen  for  x  from  which  to  reckon 

Aa; 

the  increments.     The  values  of  Ax  are  arbitrarily  assumed. 
^y=.y?  —  2x-\-^. 


a? 

y 

Aa, 

Ay 

Aa? 

2 

1 

3 

1.75 

1 

.75 

.75 

2.5 

1.3125 

.5 

.3125 

.625 

2.1 

1.0525 

.1 

.0525 

.525 

2.01 

1.005025 

.01 

.005025 

.5025 

2.001- 

1.00050025 

.001 

.00050025 

.50025 

2  +  Ax 

i+f-? 

Aa; 

2         4 

.a.f 

An  examination  of  this  table  shows  that  as  the  increment  in  ' 
X  is  made  smaller  and  smaller  the  corresponding  increment  in 
y  becomes  smaller  and  smaller,  and  approaches  the  limiting 
value  zero  when  Aa;  approaches  the  limiting  value  zero.     The 

ratio  — ^,  however,  does  not  approach  zero,  but  approaches  the 

t^x 

limiting  value  .5  when  Aa;  approaches  the  limiting  value  0. 


SLOPES 


155 


Fig.  123. 


127.  Slope  of  the  curve  at  any  point.  Look  now  at  the  geo- 
metric meaning  of  these  facts.     If  P  and  P'  denote  the  points 

(2, 1)  and  (2  -\- ^x,l  +  A?/) 
on  the  curve,  then  Aa;  and 
A?/  have  the  values  shown 
in  the  figure,  and  the  ratio 

— ^  is    the    slope    of    the 

Aa;  ^ 

secant  line  through  P  and 
P'.  As  Aa;  approaches  the 
limiting  value  zero,  the 
point  P'  moves  along  the 
curve  to  the  limiting  posi- 
tion P,  and  the  secant  line  through  P  and  P'  turns  about  P  to 
the  limiting  position  defined  to  be  the  tangent  to  the  curve  at 
P.     Hence  the  slope  of  the  tangent  line  at  (2,  1)  is  .5. 

Definition.  The  slope  of  a  tangent  to  a  curve  at  any  point 
is  called  the  slope  of  the  curve  at  that  point. 

The  method  here  employed  is  a  general  one.  By  it  one  can 
compute  the  slope  of  the  curve  at  any  point.  The  table  of 
values  need  not  be  computed,  as  in  the  preceding  article,  for 
this  purpose. 

E.g.  to  find  the  slope  of  the  curve  at  the  point  where  a;  =  3 
one  may  proceed  as  follows : 

Substitute  a;  =  3  in  the  equation ;  then  2/  =  J- 

Take  a  point  on  the  curve  near  P(3,  ^).  It  may  be  repre- 
sented by  P'  (3  +  Aa;,  -J  +  A?/). 

Since  this  point  is  on  the  curve,  its  coordinates  must  satisfy 
the  equation  of  the  curve. 

.-.  4(J  +  A2/)  =  (3  +  Aa^)2-  2(3  +  Ax)4-4, 
or  y4-4A2/  =  ^  +  6Aa!-f-A«^  — ^  — 2Aa;-|-^, 

or  4  A^/  =  4  Ax  -f-  Aa; . 

^  —  1    I  A^. 
*  *  Aa;  4 


156 


ANALYTIC  GEOMETRY 


As  Ax  approaches  the  limiting  value  zero,  i.e.  as  P'  moves 

along  the  curve  to  coincide  with  P,  the  ratio  — ^  approaches 

Ax 
the  limiting  value  1,  which  is,  therefore,  the  slope  of  the  tan- 
gent line  to  the  curve  at  the  point  (3,  J). 

EXERCISE  XXXI 

1.  Compute  the  value  of  Ay  when  Ax  =  .01  for  x  =  .5,  1,  10,  respec- 
tively, iny  =  x^. 

2.  Compute  the  slope  of  the  curve  4  y  =  x^  —  2  x  +  4  at  the  points 
where  x=—  1,  x  =  0,  x  =  4. 

3.  Find  the  slope  of  the  curve  8  y  =  x^  +  1  at  the  points  where  x  =  1, 
3,  0,-2. 

4.  Write  the  equation  of  the  tangent  line  to  the  curve  of  eq.  (1),  Art. 
126,  at  the  point  (3,  |). 


128.   Equation  of  the  tangent  to  a  curve  at  any  point.     As  a 

second  example  let  it  be  required  to  find  the  slope  of  the  tan- 
gent line  to  the  curve 

40^  +  2/^=4  (1) 
at  any  point  (xq,  y^)  on 
the  curve,  and  the  equa- 
tion of  the  tangent  line 
at  that  point. 

Let  P(xq,  2/o)  be  any 
point  of  the  curve  and  Q 
(xq  4-  Ax,  2/o  +  Ay)  a  point 
of  the  curve  near  P.  For 
convenience  Ax  is  taken 
positive,  and  then  Ay 
will  be  positive  or  nega- 
tive according  as  the 
curve  rises  or  falls  to- 
ward the  right  from  P. 

In  the  figure,  for  either  position  shown,  PM=  Ax,  MQ  =  Ay. 


^ 

/ 

Q 

/ 

r 

\ 

M 

r 

M 

\ 

I 

/ 

\ 

X 

A 

0 

1 

\ 

i 

/ 

Q 

\ 

V 

^ 

p 

M 

TANGENTS  AND  NORMALS  157 

Then  — ^  =  slope  of  the  secant  line  PQ,  and  hence  the  limit- 

ing  value  of  — ^,  as  Aa;  approaches  the  limiting  value  zero,  is 

the  slope  of  the  tangent  line  to  the  curve  at  {xq,  y^. 

Since  (a^o,  ?/o)  and  {xq  +  Ax,  y^  -\-  Ay)  are  points  on  the  curve, 
the  coordinates  must  satisfy  eq.  (1). 

.-.  4«o^  +  2//  =  4,  (2) 

and  4(a-o  +  Ax)2-f-(yo  +  A2/)2=4.  (3) 

Expanding  eq.  (3)  and  subtracting  the  corresponding  mem- 
bers of  eq.  (2),  there  results 

SxoAx  +  4:Kx-\-2  y^Ay  +  A^^  =  0.  (4) 

Every  term  of  this  equation  contains  either  Ay  or  Aa;  as  a  fac- 
tor. Take  to  the  right  member  all  the  terms  containing  Aa; 
and  factor  the  two  members  of  the  equation.     Then 

Ay (2  2/0  +  A2/)  =  -  Ax  (8  ;Bo  +  4  Aa;), 
or  A^^_8^_+4A^^  ,^. 

Aa;  2  2/0  +  Ai/ 

As  Q  moves  along  the  curve  to  the  limiting  position  P,  both 
Aa;  and  Ay  approach  the  limiting  value  zero,  and  the  right 
member  of  eq.  (o)  approaches  the  limiting  value, 

2/0 

Hence is  the  slope  of  the  tangent  line  to  the  curve  at 

2/0 

{^0,  2/o). 

Since  the  equation  of  a  line  of  slope  m  through  (a^o,  2/0)  is 

y-y^  =  m(x-XQ), 

therefore  the  equation  of  the  tangent  line  to  the  curve  at 
(a^o,  2/0)  is 

y-yo  =  — ^(x-xo).  (6) 

2/0 


158  ANALYTIC  GEOMETRY 

This  equation  may  be  put  into  a  simpler  form  as  follows : 
Clear  of  fractions : 

y^y  -  2/0^  =  -  4  aroo;  -h  4  x^^, 
or  4«oaJ  +  2/o2/  =  4a;o2  +  ?V- 

The  right  member  of  this  equation  is,  by  eq.  (2),  equal  to  4. 

Therefore  4  X(flc  -f-  ?/o2/  =  4  (7) 

is  the  equation  of  the  tangent  to 

4a;2  +  2/^  =  4, 
at  («o,  2/0). 

Since  in  eq.  (7)  (xq,  2/0)  may  be  any  point  on  the  curve,  the 
equation  of  the  tangent  line  at  any  particular  point  may  be 
written  by  substituting  for  a^o  and  2/0  the  coordinates  of  that 
point. 

Thus  the  tangent  at  (^,  V3),  which  is  a  point  on  the  curve, 

is  2x+V32/  =  4. 

The  student  must  not  fail  to  recognize  the  fact  that  in  eq. 
(7)  Xq  and  2/0  are  the  coordinates  of  a  fixed  point,  the  point  of 
tangency,  and  that  x  and  y  are  the  variable  coordinates  of  any 
point  on  the  tangent  line. 

129.  The  normal.  The  normal  to  a  curve  at  any  point  is 
the  line  perpendicular  to  the  tangent  at  that  point. 

Since  its  slope  is  the  negative  reciprocal  of  the  slope  of  the 
tangent,  the  equation  of  the  normal  to  the  curve  of  the  pre- 
ceding article  at  (xq,  y^  is 

2/-2/o  =  -r^(a5-a;o)- 

EXERCISE  XXXn 

Find  the  equations  of  tangents  and  normals  to  the  following  curves  at 
the  points  assigned.  Check  graphically  by  plotting  the  curves  and  the 
lines  whose  equations  are  found. 

1.  ?/2  =  4a;at  (1,  -2). 

2.  a;2  +  2/2  ^  25  at  (-  3,  4). 


DERIVATIVES  159 

3.  2/2  =  a;3  at  (x^,  yo)  ;  at  (0,  0),  (1,  1),  (4,  8). 

4:.  y  =  mx  +  &  at  (Xq,  yo). 

5.  y  =  x^-\-4:X— bsit  the  points  where  the  curve  crosses  the  a;-axis. 

6.  x^-y^=  16  at  (5,3). 

7.  a;y  =  8at  (2,  4). 

8.  At  what  angles  does  the  line  y  =  3x-\-2  cut  the  parabola 
y  =  x^  +  x—  6?  (By  the  angle  between  two  curves  is  meant  the 
angle  between  their  tangents  at  the  point  of  intersection.) 

9.  Find  the  angles  at  which  x^  4-  y'^  =  25  and  ix^  =  9y  intersect. 

10.  Find  the  point  on  the  curve  of  example  5  where  the  slope  is  zero. 

11.  Find  the  point- on  the  curve  y=—  x^  —  Sx-\-2  where  the  slope  is 
zero.     Find  also  the  point  of  the  curve  where  the  slope  is  1.     Where  2. 

130,  Tangent  equations  for  reference.  By  the  method  used 
in  Art.  128,  the  student  can  show  that  the  following  are  the 
equations  of  the  tangents  to  the  given  curves  at  the  point 
(^0, 2/o). 


Equation  of  Curve 

Equation  of  Tangent 

y,^  =  2poo. 

^<pc=p{y  +  yo^' 

a?2     2/2 
a2      h^ 

a'        62 

nay  =  c. 

i^oV  +  2/oi»  =  2c. 

The  student   can  more   easily  derive  these  equations  after 
reading  the  remainder  of  this  chapter. 

DERIVATIVES.    FORMULAS  OF  DIFFERENTIATION 


131.  Definitions  and  Notation.  In  the  preceding  articles  the 
limiting  value  of  — ^  as  Ax  approached  the  limiting  value  0  at 
any  point  of  the  curve  was  found  to  represent  the  slope  of  the 


160 


ANALYTIC  GEOMETRY 


tangent  to  the  curve  at  that  point.     This  limiting  value  of  — ^ 

Ax 

is  of  great  importance  in  much  advanced  mathematical  work,  as 
well  as  for  the  study  of  curves.  It  is  therefore  worth  while 
to  assign  a  special  name  to  this  limiting  value,  and  to  develop 
short  methods  for  computing  it  in  given  cases. 

Definition.  Given  a  function  y,  of  a  variable  x,  and  a 
pair  of  corresponding  values  of  x  and  y  ;  if  then  an  increment 
Ax  be  given  to  x,   bringing  about  an  increment  Ay  in  y,  the 

limiting  value  of  — ,  as  Ax  approaches  the  limiting  value  zero, 

Ax 

is  called  the  derivative  of  y  with  respect  to  a?  for  that  value  of  x. 
Notation.     The  symbol  -J^  is  used  to  denote  the  derivative 


dx 


of  y  with  respect  to  x.     The  symbol  -^ 

(XX 


means  the  value  of 


that  derivative  for  the  value  x^  of  x. 

Thus  in  Art.  128,  in  the  equation  4  a;-  +  /  =  4, 


dy 
dx 

__      4a;o 
2/0 

dy 
dx 

_        2 

132.  Geometric  meaning  of  the  derivative.     A  function  y,  of 

a  variable  x,  may  be  represented 
graphically  by  a  curve. 

Let  Xq  and  2/0  be  a  pair  of  cor- 
responding values  of  x  and  y. 
They  are  then  the  coordinates 
of  some  point  on  the  curve.  If 
an  increment  Ax  be  given  to  x, 

—  then  y  takes  an  increment  Ay, 

as    illustrated    in    the    figure, 
Fig.  125.  where  FM  =  Ax,  MQ  =  Ay. 


DERIVATIVES  161 

Then  -^  is  the  slope  of   the  secant  line  through  P(xq,  y^ 

Ax 

and  Q  (x^  +  Aa;,  yo  +  A?/). 

Let  Q  move  along  the  curve  to  the  limiting  position  P ;  Aaj 
and  Ay  both  approach  the  limit  0,  and  the  secant  line  ap- 
proaches the  limiting  position  of  the  tangent  to  the  curve  at  P. 

Hence  the  limiting  value  of  — ^,  as  Ax  approaches  the  limit 

Ax 

0,  is  the  slope  of  the  tangent  to  the  curve  at  P{xq,  y^).     There- 
fore, 


doc 


=  the  slope  of  the  tangent  to  the  curve  at  (a?o,  y^. 

The  process  of  obtaining  the  derivative  is  called  differen- 
tiation. 


133.  Continuity  of  functions.  In  the  foregoing  it  was  as- 
sumed that  2/  is  a  single-valued,  continuous  function  of  x  for 
all  values  of  x  under  discussion.  The  meaning  of  this  is  ex- 
plained in  the  following  definition. 

Definition.  A  function  y,  of  a  variable  x,  is  said  to  be 
a  single-valued  and  continuous  function  for  all  values  of  x 
within  an  interval,  if  for  each  value  of  x  in  that  interval  there 
is  a  single,  real,  finite  value  of  y,  and  if  y  changes  gradually 
as  X  changes  gradually,  i.e.  such  that  the  change  in  y  caused 
by  a  change  in  x,  anywhere  within  the  interval,  can  be  made 
small  at  will  by  making  the  change  in  x  small  enough. 

If  y  becomes  infinite  as  x  approaches  a  certain  value  as  a 
limit,  y  is  said  to  have  an  infinite  discontinuity  at  that  value 
of  X. 

If,  as  X  passes  through  a  certain  value,  y  changes  suddenly 
from  one  finite  value  to  another,  y  is  said  to  have  a  finite 
discontinuity  at  that  value  of  x. 

Example  1.     In  ?/  = as  x  approaches  the  limit  2  from 

X —  2 


162  ANALYTIC  GEOMETRY 

either  side,  y  increases  indefinitely  in  numerical  value.    Hence 
y  has  an  infinite  discontinuity  at  a;  =  2. 

2*  —  1 
Example  2.   In  y  =  —^ if  x  is  negative,  but  numerically 

J  2^  +  1 

very  small,  2*  is  very  small  and  y  is  very  near  —  1.     Again, 

_i 

1  — 2  ^ 
y  may  be  written  y  = ^,  from  which  it  is  evident  that  y  is 

1-f  2~^ 
very  near  1  when  x  is  positive  and  very  small.     Hence  as  x 
passes  through  0  from  negative  to  positive,  y  changes  suddenly 
from  —1  to  +1.     Therefore  y  has  a  finite  discontinuity  at 
x  =  0. 

All,  or  nearly  all,  of  the  functions  with  which  the  student 
ordinarily  deals  are  either  continuous  or  have  infinite  discon^ 
tinuities  at  definite  points  separated  by  finite  intervals,  and  it 
will  be  assumed  in  what  follows  that  the  functions  dealt  with 
are  finite  and  continuous  for  the  values  of  the  variable  con- 
sidered. 

134.  Formulas.  In  the  following  articles  some  general  for- 
mulas of  differentiation  will  be  developed  which  will  shorten 
the  work  of  differentiation  in  certain  cases. 

135.  Derivative  of  a  constant.  The  derivative  of  a  constant 
is  zero : 

^  =  0. 
due 

Proof.  Let  C  be  any  constant.  Since  C  does  not  change 
as  X  changes  by  any  amount  Aa;,  the  increment  in  C  is  zero : 
i.e.  AO=0. 

^  =  0. 

Ao; 


DERIVATIVES     •  163 

AC 

Therefore  the  limiting  value  of  —  is  zero, 

Ax 

dC     f. 
or  —-  =  0. 

ax 

This  may  also  be  seen  geometrically  by  letting  y  =  C.  This 
is  the  equation  of  a  straight  line  parallel  to  the  a^axis.     The 

value  of  -^  at  any  point  of  this  line  is  zero.     (Art.  132.) 

Hence  ^  =  0,  or  since  y=C,  —  =  0. 
dx  dx 

136.  Derivative  of  a  variable  with  respect  to  itself.     The 

derivative  of  a  variable  with  respect  to  itself  is  1 : 

dx 

Proof.  -^  =  1. 

Aa; 

Therefore  the  limiting  value  of  —  is  1. 

i^X 

dx  _  ^ 
dx 
The  student  should  illustrate  this  geometrically. 

137.  Derivative  of  a  constant  times  a  function.  The  deriva- 
tive of  a  constant  times  a  function  is  equal  to  the  constant 
times  the  derivative  of  the  function : 

d(Cu)  _  ^du 
d3c  dx 

where  C  is  any  constant  and  u  is  any  function  of  x. 

Proof.    Let  y  =  Cu. 

Let  X  take  a  particular  value  Xq.  Then  u  and  y  take  corre- 
sponding values  Uq  and  y^,  such  that 

2/o  =  Cuq. 


164 


ANALYTIC  GEOMETRY 


Let  X  take  an  increment  Aa; ;    then  u  and  y  take  increments 
Aw  and  Ai/  such  that 

yo-\-  /^y=  C  (uq  +  ^u). 
By  subtraction,  ^y  =  G  >  Au. 

Am 


Divide  by  Aa; ; 


^=0 

Ax  Ax 


As  Aa;  approaches  the  limit  0,  — ^  and  —  approach  the  limits 


dy 
dx 

~»0 

.       dy 
dx 

Since  Xq  is  any  value  of  x,  then 

dy  _  ^1  dxi 
dx  ~      dx 

or,  since  y  =  Cu, 

c 

Cu)  _  p  du 
Ix              dx 

138.  Derivative  of  a  sum.  The  derivative  of  a  sum  of  func- 
tions with  respect  to  any  variable  is  equal  to  the  sum  of  the 
derivatives  of  the  functions  with  respect  to  that  variable : 

^^u  +  v  +  w-)=^p  +  ^  +  ^  +  -. 
doc  doc      doc      due 

Proof.  (For  two  functions.)  Let  ii  and  v  be  two  functions 
of  X. 

Let  y  =  u  -{-  V. 

Let  X  =  Xq,  then  2/0  =  Uq  -\-  Vq. 

Let  x  =  Xq  -}-  Ax,  then 

yQ  +  Ay  =  Uq  4-  Am  +  Vo  +  Av. 
Subtracting,  Ay  =  Au  +  Av. 


or 


DERIVATIVES 

Divide  by  Ax,               At,  ^  A»  ^  A« 

Ax       Ax       Ax 

Let  Ax  approach  the  limit  0 ; 

dy 
dx 

du 

^r  dx 

dv 

dx                     dx 

dx 

165 


A  similar  proof  holds  for  any  number  of  functions. 


139.  Derivative  of  a  product.  The  derivative  of  the  product 
of  two  functions  is  equal  to  the  sum  of  the  products  of  each 
function  times  the  derivative  of  the  other : 


d(uv) 
dx 


doc         doc 


Proof.     Let  u  and  v  be  any  two  functions  of  x. 
Let  y  =  uv. 

Let  X  =  Xq,  then  2/0  =  ^o'^o- 

Let  x  =  Xq  -\-  Ax,  then 

yo  +  Ay  =  (uo  +  A?^)  (vq  +  Av). 
Subtracting,  Ay  =  (uq  -f  Au)  (vq  +  Av)  —  Uq  Vq 

=  Uq  Av  +  Vq  Au  -\-  Au  •  Av. 

Dividing  by  Aic,  J 

Ax 


Av   ,        Au    ,    Au     J. 
Uo—-  +  '^0-7-  +  -—  •  Av. 

Ax  Ax       Ax 


Let  Ax  approach  the  limit  0;  then  Au,  Av,  and  Ay  each 

approaches  the  limit  0,  and  the  limiting  values  -^,  — ,  and 

Av  .  .  ^^     ^^ 

^^  are,  respectively,  the  derivatives  of  y,  u,  and  v  with  respect 

to  x  for  the  value  Xq. 


dy 
dx 


=  Uq 


dv 
dx 


+  V(i 


.  du 
dx 


0, 


166  ANALYTIC  GEOMETRY 

or,  since  Xq  is  any  value  of  x,  and  y  =  uv, 
d(uv)         dv  .      du 
dx  dx        dx 

In  a  similar  way  a  formula  may  be  derived  for  tlie  derivative 
of  the  product  of  three  or  more  functions.  However,  one  may 
make  use  of  the  formula  just  proved  to  obtain  the  derivative 
of  the  product  of  more  than  two  functions.     Thus, 


d(uvw)         d(vw)  ,  ,     .du 
dx  dx  dx 


[dw  ,      dv~\ 
V \-w  — 
dx         dxj 


,        du 
dx 


dw  ,        dv  .        du 

=  uv f-  uw \-vw • 

dx  dx  dx 

140.  Derivative  of  a  quotient.  The  derivative  of  the  quo- 
tient of  two  functions  is  equal  to  the  denominator  times  the 
derivative  of  the  numerater,  minus  the  numerator  times  the 
derivative  of  the  denominator,  divided  by  the  square  of  the  de- 
nominator: ^du_^dv 

€lQC  due 


d  fu\        (Ix 

dx\vj~  v'^ 


u 
Proof.     Let  y  =  -i 

V 

then,  vy  =  u. 

Differentiating,  using  the  formula  of  the  preceding  article, 

dy  ,     dv     du 

v-^-{-y  —  =  — -  • 

dx        dx     dx 

du        dv 

Solving  for  ^,  f^dc^Jj^, 

dx  dx  V 

Eeplacing  y  by  -, 

V 

du        dv 

V u  — 

d  fu\        dx        dx 


dx\v)  v^ 


DERIVATIVES  167 

141.  Derivative  of  the  power  of  a  function.  The  derivative 
of  the  nth  power  of  a  function  is  equal  to  n  times  the  function 
to  the  power  w  —  1,  times  the  derivative  of  the  function : 

^^^^  =  lii***-!  — ,  where  n  is  constant. 
dx^  doc 

Proof.     (1)  n  a  positive  integer. 

Let  y  =  w". 

Let  X  =  Xq,  then  y^  =  Uq\ 

Let  x  =  Xq  +  Ax,  then  yQ-\-Ay=  (uq  +  Aw)'*. 

Expanding  (uQ-\-Auy  by  the  binomial  theorem  and  sub- 
tracting, 

Ay  =  niiQ^'-^Au  -f  ^  ^f  ~^^  i^o""'^  •  ^+  •••  +  Aw". 
1  •  ^ 

Every  term  on  the  right  after  the  first  contains  Au  to  a  power 

higher  than  the  first.     Set  out  the  factor  Au  and  divide  both 

members  by  Aaj : 


Ax     [_  2 


-\-Au       P-- 
Ax 


Now  as  Aa;  approaches  the  limit  0,  so  do  Au  and  Ay.     The 
limiting  value  of  the  quantity  in  the  parenthesis  is  therefore 


nV'. 

dy 
***  dx 

or 

± 

(y^^r^^n-.du^ 
dx                 dx 

(2)  n  a  negative  integer.     Let  n  =  —  my  where  m  is  a  posi- 
tive integer. 

Let  y  =  u'*  =  u~"'  =  —  • 

^  u"^ 

Differentiate,   using  the   formula  for   the   derivative  of   a 
quotient,  «  ^  _  i  dju"") 

dy  _      dx  dx 

dgo  ~~  w^** 


168  ANALYTIC  GEOMETRY 

But  —  =  0,  by  Art.  135,  and  since  m  is  a  positive  integer, 
dx 

^L(<l  =  mu'^-'—,  by  part  (1)  of  this  article. 

UX  (tX 

dy  __  —  mu'^~^  du 
dx  u^""      dx 

-m-l  ^^ 

dx 

„_i  du 
=  nw"  ^  — ,  since  n  =  —  m. 
dx 

(3)  n  a  rational  fraction.     Suppose  n  =  ^,  where  79  and  q  are 

integers,  either  positive  or  negative. 

p 
Let  y  =  u^=u^. 

Raise  both  members  of  this  equation  to  the  gth  power; 

Since  both  p  and  q  are  integers,  the  formula  of  this  article 
may  be  applied. 

.-.  qy^ 


Now 


dx' 

=  p?/P" 

dx' 

dy. 

_pu^ 

-^du 

dx 

QV 

-1  dx' 

p 

P-^ 

y'- 

=  (u^) 

i«-i  = 

u     \ 

, 

dy^ 

p  uP- 

■^du 

dx 

q    P- 

^dx 

u 

1 

p    I. 

-\du 

f»' 

dx 
1  du 

ww~- 

1 
dx' 

DERIVATIVES  169 

Hence  -^—^  =  nu*"-^  —  if  n  is  an  integer  or  the  ratio  of  two 
dx  dx 

integers. 

The  proof  can  be  extended  to  inchide  irrational  values  of  n, 

such  as  v2,  TT,  etc.,  but  it  is  not  sufficiently  elementary  to  be 

given  here. 

142.    Summary.     The  above  formulas  are  here  collected  and 
numbered  for  convenience  of  reference. 

I. 

II. 

III. 

doc  doc     doc     doc 

y  d(UV)^^dV^^^dM 

doc  due        doc 


dC 
doc 

-O. 

doc 
doc 

1. 

d(Cu)  _ 
doc 

-C 

du 
doc 

VI.  a(n^   V 


du     ^dv 
doc        doc 


doc 


yjj  d(un)  ^^^n-ldM 

doc  doc 

143.     Ulustrations.     Example  1.     To  find  the  derivative  of 
y?  -\-^x^-\-o  with  respect  to  x. 

A(a^  +  3^  +  5)=^+<f^  +  |^,  by  IV, 

dx  dx  dx         dx 

=  3iB2^  +  3  .  2a^^  +  0,  by  I,  III,  VII, 
dx  dx 

=  3  a^  4-  6  a;,  by  II. 

Example  2.     Given  z  =  4.f-\-  V^^  4- 1 ;  to  find  ^. 

(XL 

dZ^djAf)        djf+l)--  ,        jy 

dt         dt  dt       '  "^       ' 


170  ANALYTIC  GEOMETRY 


=  4  .  3  «2  .  -  +1 0'  +  0"^  ^i?^±i),    by  III  and  VII, 
=  12  f  +  ^  if  +  ir\2  1 1  +  0),  by  II  and  IV, 

=  12^2.^ 


■y/e  +  1 

Example  3.     GiYenpv  =  4;  to  find  t"- 

Differentiate  both  members  of  the  equation  with  respect  to  v, 

.d(pv)  ^cl(4:) 
dv         dv  * 

dv  ,      dp     ^ 

or  p 1-^-^  =  0, 

dv        dv 

or  p^v^  =  0. 

dv 

•    ^_  _^ 
'  '  dv         v' 

Example  4.     Given  the  ellipse  4  a;^  +  2/^  =  16 ;  to  find  the 
slope  of  the  tangent  line  at  (1,  2V3). 

The  slope  of  the  tangent  line  required  is  the  value  of  -^  at 

dx 

the  point  (1,  2V3).     (Art.  132.) 

Differentiating  both  members  of  the  equation  with  respect  too?, 

dx^       ^^^        dx 
,    Q     dx  ,  c    dy      /^ 
dx  dx 

.  ^_  _i£ 
'  '  dx  y  ' 

o 

Hence  the  slope  of  the  tangent  at  (1,  2V3)  is —  • 

V  3 


DERIVATIVES  171 

The  student  should  draw  the  ellipse  and  the  line  through 

(1,  2V3)  with  slope  -~ 
V3 

EXERCISE  XXXIII 

Find  the  derivative  of  each  of  the  following  functions  with  respect  to  its 
variable.  The  quantities  a,  &,  c,  m,  n,  are  constants.  All  other  letters 
represent  variables. 

2.  y  =  ax''  +  b  +  -' 

3.  y  =  Vx^  +  a^. 

4.  s=^-±^. 

t  —  a 


5.   y  =  xVx^  +  1. 


7. 

Z  =  t^-  «-2. 

8. 

-V^l- 

9. 

y  =  (x  +  a)"(a;+6)"'. 

10. 

11. 

y=(ax^  +  b)\ 

12. 

H^S-    ' 

6.    q  =  VW-cfi^i. 

Find  the  equations  of  the  tangents  to  the  following  curves  at  the  given 
points.     Check  by  drawing  the  curves  and  the  lines. 

13.  y  =  wx  +  6  at  (xo,  2/o).  18.    xy  =  8  at  (2,  4). 

14.  y  =  4  x2  at  (1,  4).  19.   4x2  +  16  ?/2  :^  16  at  (VS,  i). 

15.  y2  ^  4  x  at  (1,  2).  20.   y  =       ^        at  (2,  1). 

(x—  1) 

16.  x2  +  i/2^25at(-4,  3).  21.   y  =  ax2  +  6x  +  c  at  (xo,  ^o). 

17.  x2  -  y2  =  9  at  (5,  4).  22.   x  =  ay2  +  6y  +  c  at  (xq,  yo) • 

23.  Carefully  construct  the  curve  pv  =  4,  and  by  drawing  tangents 
(approximately)  at  various  points  and  measuring  their  slopes,  verify  the 

result  found  in  example  3,  Art.  143,  viz.  ii?  =  — "• 

'         dv         V 

24.  Derive  the  equations  of  the  tangents  to  the  curves  of  Art.  130. 

25.  Show  that  the  equation  of  the  tangent  to  a3i^^-[-hy'^+cx-{-dy+e=0 
at  (xo,  yo)  is 

axox  +  byoy  +  ^(x  +  Xo)  -\-^(y  +  yo)+ e  =  0. 

26.  Show  that  the  equation  of  the  tangent  to  y  =  «'  at  (xo,  2/o)  is 


172  ANALYTIC  GEOMETRY 

27.  Show  that  the  equation  of  the  tangent  to  y  =  ax^  at  (xq,  yo)  is 

^—'-^ ^^  =  axo'*-^x. 

n 

28.  Show  that  the  equation  of  the  tangent  to 

ax^  +  bxy  +  c'y^  +  dx-{-ey  +/=  0  at  (xo,  yo)  is 
axox  +  -  (xoy  +  Vox)  +  cyoy  +  -  (a;  +  a^o)  +  ^  («/  +  l/o)  +/=  0. 

144.  Limit  of  the  ratio  of  a  circular  arc  to  its  chord. 

In  Fig.  126  let  BD  and  AD  be  tangents  drawn  at  the  ends  of 
the  circular  arc  AB.     Then,  since  the  arc  of  a  circle  is  greater 

than  its  chord  and  less  than  any  line 
B         which  envelops  it  and  has  the  same 
extremities, 

chord  AB  <  arc  AB<2  BD. 
^        arc  AB        BD 
***         chovd  AB     MB' 

Now  let  the  point  A  move  along 
the  circle  to  the  limiting  position  B. 
The  line  through  A  and  B  approaches  the  limiting  position  as 
tangent  at  B,  and  hence  the  angle  MBD  approaches  the  limit  0. 

Hence   ,  which  is  equal  to  sec  MBD,  approaches  the 

limit  1. 

Therefore  the  ratio  — ^ approaches  the  limit  1  as  the 

chord  ^IB    ^^ 

arc  approaches  the  limit  0;  for  it  lies  between  1  and  a  quan- 
tity whose  limit  is  1. 

145.  Circular  or  radian  measure  of  an  angle.     The  radian  is 

defined  to  be  the  angle  at  the  center  of  a  circle  whose  arc  is 
equal  in  length  to  the  radius.  Hence  if  the  length  of  an  arc 
of  a  circle  be  divided  by  the  length  of  the  radius,  the  quotient 
is  the  number  of  radians  in  the  angle  subtended  at  the  center 


Fig.  126. 


DERIVATIVES 


173 


by  the  given  arc,  or 


arc 


radius 


angle  (in  radians). 


Hence  in  a  circle  of  radius  1,  "  the  arc  equals  the  angle." 
That  is,  the  number  of  linear  units  in  the  arc  is  equal  to  the 
number  of  radians  in  the  subtended  angle  at  the  center. 


146.    Limit  of 


In  Art.  144  if  the  circle  has  a  radius 


sine 

equal  to  1,   and  if  the  angle  MOB  is  called  6,   then  chord 
AB  =  2  sin  0,  and  arc  ^4jB  =  2  6. 

arc  AB   ^    e 
chord  AB     sin  0 

Q 

Therefore  the  ratio  — — -  approaches  the  limit  1  when  0  ap- 


sin^ 


proaches  the  limit  0. 


147.    Derivative  of  the  sine. 

Let  y  =  sin  u,  where  u  is  sl  function  of  x. 

In  a  circle  of  radius  1,  let  AOP  be  an 
angle  at  the  center  whose  measure  in  radians 
is  u.     (Fig.  127.) 

Then  MP  =  sin  it.     .-.  MP  =  y. 

Let  X  take  an  increment  Ax,  bringing 
about  an  increment  Au  in  u,  represented  by 
the  angle  POQ. 

Then  arc  PQ  =  Au,  and  SQ  =  Ay. 

In  triangle  P/SQ,  ' 

Ay  =  chord  PQ  -  sin  SPQ. 
chord  PQ 
arcPQ 
or,  since  arc  PQ  =  Au, 
Ay 


Fig.  127. 


^=smjSPQ 

Ax 


arcPQ 

Ax     ' 


Ax 


=  sin  SPQ 


chord  PQ 

SLTCPQ 


Au 
Ax 


174  ANALYTIC  GEOMETRY 

Now  as  Ax  approaches  the  limit  0  so  also  do  Aw  and  Ai/; 
the  line  through  P  and  Q  approaches  the  limiting  position  of 

the  tangent  at  P,  and  hence  the  limiting  value  of  SPQ  is  -  —  w. 

2 

Also  the  limiting  value  of  ^^^^^^^  is  1. 
^  arcPQ 

dy       .    fir       \du 
.'.  -^  z=  sm  (  —  u  ] — , 

dx  \2       Jdx' 

die  dx 

148.  Derivative  of  the  cosine.     In  Fig.  127  let 

Z  =  cos  Uj 

then  0M=  z,  NM=  -  A2. 

.-.   -  A2;=  cos  >SPQ  .  chord  PQ. 

.-.  =^  =  Go^SPq  .  ^1^2£dPQ  .    ^, 
Aa;  arc  PQ         Ax 

Therefore,  letting  Aa;  approach  the  limit  0, 
_dz^_        /tt  _    \  ^ 
dx  \2        Jdx' 

dx^  dx 

149.  Derivatives  of  sine  and  cosine  of  an  angle  not  in  the 
first  quadrant.  The  foregoing  proofs  have  assumed  the  angle 
to  be  in  the  first  quadrant.  Proofs  could  as  easily  be  given  for 
the  other  quadrants,  or  they  may  be  made  to  depend  upon  those 
above. 

E.g.  to  find  — ^  for  a  value  of  u  in  the  second  quadrant. 

dx 

Let  w=5  + v;  then  sin  w=  cos-y,  cosw=—  sinv,  and  —  =  — . 
2        '  '  '         dx     dx 

c2(sin  u)      c?(cos  v)  .       dv   ^      ,.    ,    ^mq 

/.    -^^ ^  =  ^ -^  =  —  sm  v  — ,  by  Art.  148, 

dx  dx  dx 

=  008%  — ,  as  before. 
dx 


DERIVATIVES  175 

Analytic  proofs  of  the  above  formulas  which  are  independent 
of  the  size  of  the  angle  are  given  in  text-books  on  the  Calculus. 

150.     Derivative  of  the  tangent. 

Let  y  =  tan  u. 

sinw 

.*.  y  = 

cos^^ 
Differentiating  by  the  formula  for  a  quotient, 

^^^      d(sin  u)        .       d(cos  u) 

cos  u  -^ ^  —  sm  u  -^ ^ 

ay  _  dx  dx 

dx  cos^  u 

du       '       /       .      .du 

cos  u  COS  u sm  u  {—  sm  u) — 

dx  'dx 


(cos^u-{-  sin^i^) — 
dx 

= 2 * 

^^  d(tan  u)  .,    du 

151.  Derivatives  of  cotangent,  secant,  cosecant.  The  student 
can  show  that  the  following  formulas  hold : 

dicotu)  ^^o    du  dCsecu)  .         du 

~^— — -  =  —  csc^  u  — ,  -^^ ^  =  sec  u  tan  u  — , 

ax  dx       dx  dx 

c?(csc  u)  ,     du 

-^- — ^  =  —  CSC  w  cot  w  — 
dx  dx 

152.  Summary.  The  formulas  for  the  differentiation  of  the 
trigonometric  functions  are  here  collected  and  numbered  con- 
secutively with  those  of  Art.  142. 

VIII    ^^^m^  =  cost*^. 
doc  doc 

IX    ^^^^^^>  =  -sini^^. 
doc  doc 


176  ANALYTIC  GEOMETRY 


dx  doc 


XT  ^(cot^^)^_csc2^J^. 
^  ddc  doc 

XII  ^^^^^^  =  sect*  ton t*^. 
^  cix  doc 

XIII  ^i^f«^=-csci*cott.^. 
■         rfic  dx 


163.  Illustrations.  The  foregoing  formulas,  together  with 
those  of  Art.  142,  enable  one  to  find  the  derivative  of  any  alge- 
braic expression  involving  trigonometric  functions.  The  fol- 
lowing examples  will  help  to  make  this  clear. 

Example  1.         Given  y  =  sin^  2  a;;  to  find  ^• 

dx 

By  formula  VII,  ^  =  3  sin^  2  x  I^^IA. 

dx  dx 

By  VIIIandII,iii5iEM=cos  2c. ^^M 
dx  dx 

—  2  cos  2  x. 

.'.  -^  =  6  sin^  2  X  cos  2  a;. 

^^ 

Example  2.  Given  2;  =  Vl  +  2  tan^  3  s;  to  find  -77' 

By  VII,  ^=i (1+2 tan^ 3 .)-^  ^a+2tan^38) 

^        '  dt     2^  ^  dt 

By  IV,  I,  III,  VII,  and  X, 

d(l  +  2  tan^  3  s)       .  ,       o  ^2  o  „    o    ^« 

-^ — ! L  =  4  tan  3 s  •  sec^ 6s  '  6 

dt  dt 

clz      6  tan  3  s  sec^  3  g   (^s 

'  'di^-Vl+2  tSiu'Ss'dt' 

154.  Other  derivative  formulas.  Formulas  for  the  derivatives 
of  the  inverse  trigonometric  functions,  sin~^t*,  etc.;  the  logar 


DERIVATIVES  177 

rithmic  functions,  log„  u ;  and  the  exponential  functions,  a",  w", 
are  derived  in  text-books  on  the  Calculus. 

The  foregoing  formulas  will  be  sufficient  for  use  in  showing 
the  application  of  derivatives  to  the  study  of  curves,  which  is 
given  in  the  next  chapter. 

EXERCISE   XXXIV 

Find  the  derivative  of  each  of  the  following  functions  with  respect  to 
its  variable  : 

11.  y  =  sec2  X  —  csc2  x. 

12.  y=      ^ 


1. 

y  =  a  cos2  rK  +  6  sin2  x. 

2. 

y  =  4  tan3  2  x. 

3. 

z  =  sin^  t. 

4. 

s  =  cos^x  — sin^a;. 

5. 

?/  =  cos2(a»  +  6). 

6. 

sec^x, 

CSC2  X 

7. 

y  =  sec*  3  x. 

8. 

^  _   tan  2  « 

1  +  sin  « 

9. 

y  =  sin2a;\/secic. 

10. 

y  =  tan"  mx. 

Vsin  X 

13.  z  =  ^  tan3  ^  -  tan  ^  4-  ^. 

14.  y  =  X  sina;. 

15.  y  =  X  ta.nx. 

16.  y  =  (sin  x)x  —  l. 

17.  y  =  cot  4  X  CSC  4  x. 

18.  z  =  m  cot"  qx. 

19.  ?/ =  a;(sina;— cosic). 

20.  q  =  a  sin"  bt. 


CHAPTER  XII 


MAXIMA  AND   MINIMA.    DERIVATIVE   CURVES 


155.   Maximum  and  minimum  points  of  a  curve.     In  the 

discussion  that  follows  the  curves  are  supposed  to  be  such 
that  the  ordinate  is  a  single-valued,  continuous  function  of 
the  abscissa.  If  the  curve  as  a  whole  is  not  single  valued,  it 
can  be  divided  into  portions  each  of  which  is  single  valued. 

For  convenience,  such  a  curve,  or  portion  of  a  curve,  may 
be  thought  of  as  generated  by  a  point  moving  from  left  to 
right. 

If,  as  the  curve  is  so  traced,  the  generating  point  rises  to  a 
certain  position  and  then  falls,  that  position  is  called  a  maxi- 
mum point  of  the  curve,  and  the  ordinate  at  that  point  is 
called  a  maximum  ordinate.     If  the  generating  point  falls  to 

a  certain    position    and 
C  then  rises,  that  position 

is  called  a  minimum 
point  of  the  curve,  and 
the  ordinate  at  that 
point  a  minimum  ordi- 
nate. 

Thus  A,  C,  and  E  are 

maximum  points,  and  2/1, 

2/3,  and  2/5  are  maximum  ordinates,  while  B  and  D  are  minimum 

points,  and  2/2  and  y^  are  minimum  ordinates  of  the  curve  in 

Fig.  128. 

According  to  the  above  definition  a  maximum  ordinate  is 
not  necessarily  the  greatest  ordinate  of  the  curve.  The  defini- 
tion requires  only  that  a  maximum  ordinate  shall  be  greater 
than  the  ordinates  immediately  to  the  right  and  left  of  it. 

178 


Fig.  128. 


MAXIMA  AND  MINIMA  179 

At  a  maximum  point  the  curve  is  said  to  change  from  rising 
to  falling,  and  at  a  minimum  point  to  change  from  falling  to 
rising. 

156.  Determination  of  the  maximum  and  minimum  points  of 
a  curve.  The  location  of  the  maximum  and  minimum  points 
of  a  curve  whose  equation  in  rectangular  coordinates  is  known 
may  be  determined  by  use  of  the  derivative.  The  method 
employed  is  a  general  one,  but  the  solution  of  the  equations 
is  sometimes  impossible.  In  the  case  of  equations  with  nu- 
merical coefficients,  however,  an  approximate  solution  can 
always  be  obtained. 

It  was  shown  in  Art.  132  that  -^  for  any  point  of  the  curve 
is  equal  to  the  slope  of  the  tangent  to  the  curve  at  that  point. 

Then  if  -^  is  positive  for  a  given  point  of  the  curve,  the  tan- 
da; 

gent  line  at  that  point  makes  with   the  ic-axis  an  angle  less 

than  90°,  and  hence  the  curve  rises  toward  the  right  from  that 

point.     If  -^  is  negative  for  a  given  point  of  the  curve,  the 
dx 

tangent  at  that  point  makes  with  the  a>-axis  an  angle  between 

90°  and  180°,  and  hence  the  curve  falls  toward  the  right  from 

that  point. 

Of   course   this   rising  or  falling  may  continue  for  a  very 

short  distance  only. 

Figure  129  illustrates  points  of  the  curve  for  which  -^  is 
respectively  positive,  zero,  and  negative. 


dx 


T^^^X"       -^ 


dy.     ^  (Jy  dy 

ite>o  ii=°  dl<o 

Fig.  129. 


180  ANALYTIC  GEOMETRY 

It  follows  from  the  above  that  if  a  point,  in  moving  along 
the  curve   from  left  to  right,  passes  through  a   position   for 

which  -^  changes  from  positive  to  negative,  the  curve  changes 
dx 

at  that  point  from  rising  to  falling,  and  hence  that  position  is 

a  maximum  point  of  the  curve ;  while  if  a  point,  in  moving 

along  the  curve  from  left  to  right,  passes  through  a  position 

for  which  -^  changes  from  negative  to  positive,  such  a  posi- 

(XX 

tion  is  a  minimum  point  of  the  curve. 

The  derivative  -^  usually  changes  sign  by  passing  through 

ClX 

the  value  zero,  so  that  the  tangent  at  a  maximum  or  minimum 
point  of  the  curve  is  usually  parallel  to  the  avaxis.     However, 

it  may  change  sign 

by  becoming  infinite. 

^    ^         Y        I^  such  a  case   the 

J\  tangent    is    parallel 

to   the   ^/-axis   at    a 
— maximum    or   mini- 
Maximum  Points.              Minimum  Points.       mum  point.    A  point 
^^«-  1^-  of  this  kind  is  called 

a  cusp-maximum,  or  a  cusp-minimum.     (Fig.  130.) 

It  does  not  follow,  conversely,  that  if  the  tangent  at  a  given 
point  of  the  curve  is  parallel  to  one  of  the  coordinate  axes,  the 
point  is  necessarily  a  maximum  or  minimum  point.  The  curve 
may  cross  the  tangent  at  that  point.     (See  Fig.  129.) 

The  above  discussion  applies  to  only  those  parts  of  a  curve 
for  which  neither  coordinate  becomes  infinite.     It  frequently 

happens  that  as  x  passes  through  a  certain  value,  -^  changes 

sign,  but  neither  a  maximum  nor  minimum  point  of  the  curve 
corresponds  to  that  value  of  x,  because  y  there  becomes  infinite. 

157.   Illustration.      To   find  the   maximum   and   minimum 


MAXIMA  AND  MINIMA  181 

points  of  the  curve 

6y  =  2a^-3a^-S6x-12,  (1) 

Difeerentiating,      6  ^  =  6  a.-^  -  6  a;  -  36. 
dx 

...  ^  =  a^_a;_6  (2) 

dx 

=  (x-\-2)(x-3). 

From  this  it  is  seen  that  if  x  has  any  value  less  than  —  2, 

both  factors  of  -^  are  negative,  and  hence  -^  is  positive.     The 
dx  dx 

curve  therefore  rises  toward  the  right  at  all  points  for  which 

x<-2. 

If  X  is  greater  than  —  2  but  less  than  3,  one  factor  of  -^  is 

dx 

positive  and  the  other  negative,  and  hence  -^  is  negative.     The 

dx 

curve  therefore  falls  toward  the  right  for  all  values  of  x  be- 
tween —  2  and  3. 

If  a;>3,  -^  is  positive,  and  hence  the  curve  again  rises 
dx 

toward  the  right  for  all  values  of  a;  >  3. 

As  X  passes  through  —  2  from  left  to  right,  -^  changes  from 

dx 

positive  to  negative,  arid  hence  the  point  of  the  curve  for  which 
a;  ^  —  2  is  a  maximum  point ;  i.e.  (—2,  5^)  is  a  maximum  point. 

As  X  passes  through  3  from  left  to  right,  —  changes  from 

dx 

negative  to  positive,  and  hence  (3,  —  15i)  is  a  minimum  point. 

Figure  131  shows  the  curve  plotted  from  these  considerations 
and  a  few  additional  points  through  w^hich  it  passes. 

The  meaning  of  the  dotted  curve  is  explained  in  the  next 
article. 


182 


ANALYTIC  GEOMETRY 


158.   The  first  derivative  curve.     The  faqts  of  the  preceding 
article  are  clearly  brought  out  graphically  by  plotting  the  curve 

of  eq.  (2),  using  x  as  abscissa  and 


-^  as  ordinate. 
dx 


dy 


¥m.  131. 


Eor  convenience  let  -^  be  rep- 
dx  ^ 

resented  by  z.     Then  eq.  (2)    be- 
comes „  _ 

Z  =  Qr  —  X—K). 

This  curve,  being  a  parabola, 
is  easily  plotted.  It  crosses  the 
a;-axis  at  —  2  and  3,  has  its  vertex 
at  (|-,  —  -2^),  and  its  axis  parallel 
to  the  2/-axis  (Art.  81).  The  locus 
is  the  dotted  curve  in  Fig.  131. 

The  original  curve  will  be  re- 
ferred to  as  the  primitive  curve, 
and  the  curve  just  described  as 
the  first  derivative  curve. 

From  the  relations  established 
in  the  preceding  article,  it  follows 
that  for  those  values  of  x  for 
which  the  first  derivative  curve 
dy 


is  above  the  a^axis,  that  is,  2;,  or  --^,  is  positive,  the  primitive 

dx 

curve  rises  toward  the  right;  for  those  values  of  x  for  which 
the  derivative  curve  is  below  the  a^-axis,  the  primitive  curve 
falls  toward  the  right;  for  a  value  of  x  at  which  the  first  de- 
rivative curve  crosses  the  a^-axis  from  above,  in  going  from  left 
to  right,  the  slope  of  the  primitive  curve  changes  from  positive 
to  negative,  and  hence  the  primitive  curve  has  a  maximum 
point ;  and  for  a  value  of  x  at  which  the  first  derivative  curve 
crosses  the  avaxis  from  below  in  going  from  left  to  right,  the 
primitive  curve  has  a  minimum  point. 


DERIVATIVE  CURVES  183 

Moreover,  the  value  of  the  ordinate  of  the  first  derivative 
curve  gives  one  a  good  idea  of  the  rapidity  with  which  the 
primitive  curve  is  rising  or  falling.  Thus,  if  for  a  certain 
value  of  Xj  the  ordinate  of  the  first  derivative  curve  is  positive 
and  numerically  large,  the  primitive  curve  is  rising  rapidly 
toward  the  right  for  that  value  of  x ;  while  if  the  ordinate  of 
the  first  derivative  curve  is  negative  and  numerically  small, 
for  a  certain  value  of  x,  the  primitive  curve  is  falling  slowly 
toward  the  right  for  that  value  of  x. 

This  is  at  once  evident  on  remembering  that  the  ordinate  of 
the  first  derivative  curve  is  equal  to  the  slope  of  the  primitive 
curve  for  the  same  value  of  x. 

159.  Concavity.  Suppose  that  for  x  =  Xq  the  first  derivative 
curve  has  a  positive  slope. 

Let  Zq,  %,  and  z^  be  the  ordinates  of  points  on  the  derivative 
curve  for  the  values  Xq,  Xq  —  Ax,  and  Xq  +  Aa;  respectively,  and 
let  Ax  be  chosen  small  enough  so  that  Zi<.  Zq  <.  Z2* 

Then,  since  the  values  of  z  are  equal  to  the  slopes  of  the 
primitive  curve  for  the  same  values  of  x,  the  tangent  to  the 
primitive  curve  must  have  turned  counter-clockwise  as  x  in- 
creased through  Xq  from  Xq  —  Aa;  to  Xq  -\-  Ax.  This  *is  true 
whether  Zq  be  positive,  negative,  or  zero.     (See  Fig.  132.) 

Exercise  1.  In  the  curve  y  =  a^  ■}- 2  x -\- 4:  draw  the  deriva- 
tive curve,  measure  the  ordinates  at  a;  =  1^,  2,  2^,  and  draw 
the  tangents  to  the  primitive  curve  at  points  corresponding  to 
the  selected  values  of  x.  How  would  the  tangent  to  the  prim- 
itive curve  turn  as  x  increases  through  2  ?  Do  the  same  for 
aj  =  -2i.,  -2,  -11. 

*  This  is  possible  since  the  slope  of  a  curve  at  any  point  is  the  limiting 
value  of  the  slope  of  a  secant  line  through  that  point  and  a  neighboring  point 
of  the  curve.  The  secant  line,  cutting  either  to  the  right  or  left  of  the  given 
point,  can  then  be  brought  near  enough  to  the  tangent  to  have  a  positive  slope, 
since  the  slope  of  the  tangent  is  positive.  The  ordinates  of  the  curve  are 
therefore  greater  just  to  the  right  and  less  just  to  the  left  than  the  ordinate 
at  the  point  of  tangency. 


184 


ANALYTIC  GEOMETRY 


a  Derivative  Curve. 


b  Primitive  Curve. 


Fig.  132. 


Exercise  2.  Prove  that  if  the  derivative  curve  has  a  nega- 
tive slope  for  X  =  Xq,  the  tangent  to  the  primitive  curve  turns 
clockwise  as  x  increases  through  Xq. 

Exercise  3.  Illustrate  the  law  stated  in  exercise  2  by  using 
the  curve  y  —  —  x^-{-2x  —  3. 

Definitions.  If  the  tangent  to  a  curve  turns  counter- 
clockwise as  the  point  of  tangency  moves  to  the  right  through 
a  given  point,  the  curve  is  said  to  be  concave  up  at  that  point ; 
while  if  the  tangent  turns  clockwise  as  the  point  of  tangency 
moves  to  the  right  through  a  given  point,  the  curve  is  said  to 
be  concave  down  at  that  point. 

A  point  on  the  curve  where  the  curve  changes  from  concave 
up  to  concave  down,  or  vice  versa,  is  called  a  point  of  inflexion. 

As  the  point  of  tangency  passes  through  a  point  of  inflexion, 
the  tangent  line  changes  the  direction  of  rotation.  The  curve 
crosses  the  tangent  at  a  point  of  inflexion. 


Slope  increasing 
toward  the  right. 
Curve  concave  up. 


Slope  decreasing 
toward  the  right. 
Curve  concave  down. 
Fig.  133. 


Point  of  inflexion. 


DERIVATIVE  CURVES  185 

The  results  of  this  article  may  be  stated  as  follows:  For  all 
values  of  x  for  which  the  first  derivative  curve  is  rising  toward 
the  right,  the  primitive  curve  is  concave  upward  ;  for  all  values 
of  X  for  which  the  first  derivative  curve  is  falling  toward  the 
right,  the  primitive  curve  is  concave  downward ;  for  a  value 
of  X  for  which  the  first  derivative  curve  has  a  maximum  or 
minimum  point,  the  primitive  curve  has  a  point  of  inflexion. 

160.  The  second  derivative.  The  derivative  of  a  function 
of  a  variable  is  itself  a  function  of  that  variable.  This  de- 
rivative may  then  also  be  differentiated. 

Thus,  if  2/  =  2ar3  +  sin  2a;, 

^=6a;2  + 2  cos  2a;, 
dx 

and  —  (^\  =  12  a;  -  4  sin  2  a;. 


) 


dx\dx 

The  derivative,  ~ -,  is  called  the  first  derivative  of  y  with  re- 
dx 

spect  to  X,  and  — (  — )  is  called  the  second  derivative  of  y  with 

CLX  \CIXJ 

respect  to  x. 

The  symbol  ^^  is  used  to  denote  the  second  derivative  of 

y  with  respect  to  x,  thus  -^  =  ^(  ^). 
^  ^  '  dx"     dx\dxj 

Similarly,  —^  means  —  — f -^ 
doif  dx\jXx\dx 

161.  The  second  derivative  curve.  The  second  derivative  is 
related  to  the  first  derivative  in  precisely  the  same  way  as  the 
first  derivative  is  related  to  the  primitive  function.  But  it 
also  has  an  interesting  and  important  relation  to  the  primitive 
function,  now  to  be  explained. 


186  ANALYTIC  GEOMETRY 

Suppose  the  second  derivative  to  be  represented  by  a  curve, 

using  X  for  abscissa  and  -^  as  ordinate.     This  curve  is  called 
dmr 

the  second  derivative  curve. 

Then,  for  all  values  of  x  for  which  the  second  derivative 
curve  is  above  the  a7-axis,  the  primitive  curve  is  concave  up ; 
for  the  ordinate  of  the  second  derivative  curve  is  equal  to  the 
slope  of  the  first  derivative  curve  for  the  same  value  of  x,  and 
where  the  slope  of  the  first  derivative  curve  is  positive,  the 
primitive  curve  is  concave  up.     (Art.  159.) 

In  like  manner  it  is  proved  that  for  those  values  of  x  for 
which  the  second  derivative  curve  is  below  the  i»-axis,  the 
primitive  curve  is  concave  down. 

For  a  value  of  x  at  which  the  second  derivative  curve  crosses 
the  ic-axis,  the  first  derivative  curve  has  either  a  maximum  or 
minimum  point,  and  hence  the  primitive  curve  has  a  point  of 
inflexion.     (Art.  159.) 

162.  Summary.  The  results  of  the  foregoing  discussion  of 
this  chapter  may  be  summarized  as  follows : 

For  all  values  of  x  for  which  the  first  derivative  curve  is 
above  the  a;-axis,  the  primitive  curve  rises  toward  the  right ; 
for  all  values  of  x  for  which  the  first  derivative  curve  is  below 
the  a>axis,  the  primitive  curve  falls  toward  the  right;  for  a 
value  of  X  at  which  the  first  derivative  curve  crosses  the  a^axis 
from  above  in  going  from  left  to  right,  the  primitive  curve  has 
a  maximum  point ;  for  a  value  of  x  at  which  the  first  deriva- 
tive curve  crosses  the  a^axis  from  below  in  going  from  left  to 
right,  the  primitive  curve  has  a  minimum  point. 

For  all  values  of  x  for  which  the  second  derivative  curve  is 
above  the  avaxis,  the  primitive  curve  is  concave  up;  for  all 
values  of  x  for  which  the  second  derivative  curve  is  below  the 
flj-axis,  the  primitive  curve  is  concave  down ;  for  a  value  of  x 
at  which  the  second  derivative  curve  crosses  the  aj-axis,  the 
primitive  curve  has  a  point  of  inflexion. 


DERIVATIVE  CURVES 


187 


163.   Illustrations.     Example  1.     Given 


then 
and 


-^  =  cos  X, 
dx 


dx" 


—  sm  X. 


The  curves  are  shown  in  Fig.  134,  and  the  relations  established 
above  are  seen  to  hold. 


Y 

y 

\^ 

V 

y 

IfN 

>\ 

4 

y 

X 

s 

0 

^^^ 

;x' 

y 

TT 

Fig.  134. 

The  student  should  make  a  careful  study  of  the  figure. 

Example  2.     As  another  illustration,  study  the  curves  of 
Fig.  131.     The  straight  line  in  the  figure  represents  the  equa- 
tion ^'y  ^2x     1 
dx^ 

Example  3.  A  circular  cistern  is  to  be  built  to  have  a 
given  capacity ;  to  find  its  dimensions  in  order  that  the 
amount  of  lining  required  will  be  a  minimum. 

Let  H  =  depth,  D  =  diameter,  and  S  =  area  of  inner  surface. 

4 


Then 


S 


-\-TrDH. 


But 


vol.  = 


i)2  + 


=  C,  where  C  is  constant. 
40 


D 


188  ANALYTIC  GEOMETRY 

Here  S  is  expressed  as  a  function  of  the  variable  D.  From 
the  equation  it  is  at  once  evident  that  if  D  is  very  small  the 
surface  is  very  large,  and  is  again  very  large 
when  D  is  very  large ;  while  for  intermedi- 
ate values  of  D  the  surface  has  smaller 
values.  The  curve  which  represents  the 
■  equation  between  S  and  D  therefore  falls 
and  then  rises  as  D  increases  from  0,  as  in 
Fig.  135.  There  will  therefore  be  a  mini- 
mum point,  which  may  be  found  by  equating 

to  0  the  value  of  ^. 
■    t  dD 


Equating  this  expression  to  0,  and  solving  for  D, 


i.=^ 


"8(7 


The  relation  between  D  and  H  is  most  easily  obtained  by 
replacing  (7  by  ^^— —  in  the  expression  for  — ,  and  equating 

^  CHJ 

the  result  to  0.     Then 

2  4Z)^    "^' 

or  B  =  2H, 


EXERCISE  XXXV 

Sketch  the  following  curves,  first  sketching  the  first  and  second  deriva- 
tive curves.  Locate  maximum  and  minimum  points  and  points  of 
inflexion. 

1.  y  =  a;2-4rc  +  5.  /        ^\ 

5    y  =  cos  [x I  • 

2.  y=_x2-rr  +  3.  "^  \        6/ 

3.  3  2/  =  x8- 12«+ 6.  6.   6?/ =  2x3-3a;2_12x-6. 

4.  y  =  sin2x.  7.    10  y  =  2  ic^  +  9a;2  -  24  a;  +  20. 


DERIVATIVE  CURVES  189 

8.  20ij  =  x'^  -^9x'^  +  l5x-  20.         13.    ?/  =  sin  x  +  a. 

9.  y  =  x^-Sa'^x  +  b^.  14.   y  -  l=(x-  2)3. 
10.y  =  xK                                             ^^^         ,3 


11.  2/  =  a-(a  —  a:).  x  -  4 

12.  y  =  ax^  +  hx  +  c.  IQ.   Zy  =  x^ -^x'^ +  Qx  -  1. 
17.   12  y  =  3  x''  -  8  a:^  -  30  ic2  +  72  a;  +  24. 

18.  8  ?/  =  ic^  _  6  ic2  +  8  a;  +  16. 

19.  In  y  =  ax^  -{■hx  +  c^  where  a  t^  0,  show  that  there  is  a  maximum 
and  a  minimum  point  if  h  and  a  are  opposite  in  sign,  hut  that  there  is 
neither  maximum  nor  minimum  if  a  and  h  are  of  the  same  sign,  or  if 
6  =  0. 

Compare  the  curves  obtained  by  using  the  following  values  of  a,  h,  and 
c.  (1)  a  =  1,  6  =  -  3,  c  =  2  ;  (2)  a  =  1,  6  =  -  .03,  c  =  2  ;  (3)  a  =  1, 
J)  =  —  .0003,  c  =  2.  If  a  >  0,  and  a  and  c  are  held  fast  while  h  is  made 
to  approach  the  limit  0  from  the  negative  side,  what  becomes  of  the 
maximum  and  minimum  points  ?  If  6  then  becomes  positive,  how  is  the 
tangent  at  the  point  of  inflexion  affected  ? 

20.  It\  y  —  ay?  -\- hx'^  -{■  ox -\-  d  show  that  there  is  a  maximum  and  a 
minimum  point  if  62_3Qrc>0,  but  not  otherwise.  How  does  the  case 
where  h"^  —  3  ac  =  0  differ  from  that  where  &2  _  3  ^c  <  0  ? 

31.  The  equation  of  the  path  of  a  projectile,  fired  at  an  angle  a  to  the 
horizontal  with  an  initial  velocity  F,  is 

y  =  x  tan  a ^ • 

^  2  r2  cos2  a 

Find  the  maximum  height  to  which  the  projectile  rises.     Ans. ^^^  ". 

22.  Letting  B  =  the  range  on  the  horizontal  of  the  projectile  described 
in  ex.  21,  show  that  B  =  ZI^RA^. 

g 

Letting  a  vary,  plot  the  curve  which  represents  JR  as  a  function  of 

a.     For  what  value  of  a  is  i?  a  maximum  ?    Ans.    ^ . 

4 

23.  Prove  that  the  greatest  rectangle  of  a  given  perimeter  is  a  square. 

24.  A  cylindrical  tin  can,  closed  at  both  ends,  is  to  be  made  to  have  a 
certain  capacity.  Show  that  the  amount  of  tin  used  will  be  a  minimum 
when  the  height  equals  the  diameter. 

25.  Show  that  the  rectangle  of  greatest  area  that  can  be  inscribed  in  a 
circle  is  a  square. 


190  ANALYTIC  GEOMETRY 

26.  Given  that  the  strength  of  a  rectangular  beanj  of  given  length 
varies  as  the  product  of  the  breadth  and  the  square  of  the  depth,  find  the 
ratio  of  depth  to  breadth  of  the  strongest  beam  that  can  be  cut  from  a 
cylindrical  log.     Ans.  h  =  V2  •  b. 

27.  Given  that  the  deflection,  under  a  given  load,  of  a  rectangular 
beam  of  given  length,  varies  inversely  as  the  product  of  the  breadth  and 
the  cube  of  the  depth,  find  the  ratio  of  depth  to  breadth  of  the  beam  of 
least  deflection  that  can  be  cut  from  a  cylindrical  log.    Ans.  h  =  VS  •  &. 

Suggestion.    Make  the  reciprocal  of  the  deflection  a  maximum. 

28.  A  rectangular  piece  of  tin  of  vyidth  b  is  to  be  bent  up  at  the  sides 
to  form  an  open  trough  of  rectangular  cross  section.  Find  the  width  of 
the  strip  bent  up  at  each  side  vv^hen  the  carrying  capacity  is  a  maximum. 

Ans.  ^. 
4 

29.  Find  the  dimensions  of  the  greatest  right  circular  cylinder,  the 
sum  of  the  length  and  girth  of  which  is  6  ft. 

Ans.  H=2tt.,  Diam.  =  -  ft. 

IT 

30.  Find  the  dimensions  of  the  greatest  rectangular  box  of  square  base, 
the  sum  of  the  length  and  girth  of  which  is  6  ft.     Ans.  Length  =  2  ft, 

31.  Find  the  ratio  of  altitude  to  radius  of  base  of  the  conical  vessel,  of 
open  base,  which  requires  the  least  amount  of  material  for  a  given  capacity. 

Ans.  Alt.  =  V2  rad. 

32.  A  point  moves  along  a  straight  line.  At  the  time  t  its  distance 
from  a  fixed  point  of  the  line  is  s  :  at  the   time  t  +  At,  its  distance  is 

As 
s  +  As.     Then  ^  is  the  average  velocity  of  the  point  for  the  time  At. 

As 
The  limiting  value  of  ^ ,  as  At  approaches  0  as  a  limit,  is  defined  to  be 

the  velocity,  v,  at  the  time  t.    Hence  v  =  —. 

dt 
Given  s  =  16  «2,  find  the  velocity  at  any  time  t. 

33.  The  average  acceleration,  during  an  interval  of  time,  of  a  point 
moving  in  a  straight  line,  is  the  increase  in  velocity  during  that  time, 
divided  by  the  length  of  the  interval  of  time. 

Make  a  definition  for  the  acceleration  at  any  instant,  and  show  that 
the  acceleration  is 

dv    ^^^ 
dt         dt^ 
Find  the  acceleration  if  s  =  16  t"^. 


DERIVATIVE  CURVES  191 

34.  Plot  the  curves  representing  the  space,  velocity,  and  acceleration, 
in  terms  of  the  time,  if  s  =  16  t^. 

35.  Given  s  =  at^  -\-  ht  +  e,  where  a,  6,  and  c  are  constant,  show  that 
the  velocity  in  terms  of  the  time  is  represented  by  a  straight  line,  and  that 
the  acceleration  is  constant. 

36.  The  formula  for  the  space  traversed  by  a  body  projected  vertically 
upward,  with  velocity  Vq,  is 

s  z=VQt  —  16  «2  (s  in  ft.,  t  in  sees.) 

Find,  by  differentiation,  the  velocity  and  acceleration  of  a  bullet  fired 
upward  with  initial  velocity  of  1000 //s. 

Plot  the  curves  representing  space,  velocity,  and  acceleration  in  terms 
of  the  time.     How  high  does  the  bullet  go  ? 

37.  A  point  moves  back  and  forth  along  a  diameter  of  a  circle  of  radius 
a,  with  simple  harmonic  motion  (Art.  116),  making  n  complete  oscilla- 
tions per  unit  of  time.  If  s  is  the  abscissa  of  the  point  referred  to  the 
center,  and  the  point  is  at  the  end  of  the  diameter  when  «  =  0,  show  that 

s  =  a  cos(2  irnt). 
Find  also  the  velocity  and  acceleration  at  any  time,  and  plot  the  curves 
for  space,  velocity,  and  acceleration. 

38.  Since  — (x^  +  C)   is  the  same  as  — (x2\    how  many  primitive 

dx  dx 

curves  are  there  whose  first  derivative  curve  is 

-^  =  20.? 
dx 

Sketch  some  of  the  derivative  curves.  How  are  they  situated  with  refer- 
ence 'o  each  other  ?  What  is  the  equation  of  the  primitive  which  passes 
through  (2,  6)  ? 

39.  Find  the  primitives  of  which  —  =  cos  x   is  the  first  derivative 

dx 

curve. 

d^v 

40.  Find  the  primitive  of  which  Jl  =  ^  is  the  second  derivative  curve, 

and  which  passes  through  (4,  1)  with  a  slope  equal  to  3. 

d^v 

41.  Show  that  for  the  second  derivative  curve  ^  =^  ^'  ^  primitive  may 

be  obtained  which  passes  through  any  given  point  in  any  given  direction. 


CHAPTER   XIII 


THE   CONIC    SECTIONS 


164.  Definition  of  the  conic.  A  conic  section,  or  simply 
conic,  is  the  curve  of  intersection  of  the  surface  of  a  right  cir- 
cular cone  and  a  plane.  It  can  be  shown,  however,  that  the 
following  definition  is  equivalent  to  the  one  just  given. 

Definition.  A  conic  is  the  locus  of  a  point  which  moves 
m  a  plane  so  that  the  ratio  of  its  distance  from  a  fixed  point 
in  the  plane  to  its  distance  from  a  fixed  straight  line  in  the 
plane  is  constant. 

This  definition  will  be  adopted  here. 

The  fixed  point  is  called  the  focus,  the  fixed  straight  line  the 
directrix,  and  the  constant  ratio  the  eccentricity,  of  the  conic. 


165.   Construction  of  conies.     Let  F  be  the  focus,  DD'  the 
directrix,    and   e   the   eccentricity.     Let  P  be  any  point   on 

the  conic,  and  M  the  foot  of  the 
perpendicular  drawn  from  P  to 
the  directrix.      Then,  by  defini- 
tion of  the  conic, 
FP  ^ 
MP 
(The  lines  FP  and  MP  are  to  be 
counted    as    positive,    whatever 
their  direction.) 

This  suggests  the  following 
method  of  locating  points  of  the 
conic :  Through  F  draw  a  line 
FB  perpendicular  to  DD\  intersecting  DU  in  B.  Through  B 
draw   a   line   BL,   making   an   angle   6   with   BF  such   that 

192 


Fia.  136. 


THE  CONIC  SECTIONS  193 

tan  0  =  e.  Take  any  point  H  on  BF,  and  let  the  perpendicular  to 
BF  through  H  meet  BL  in  K.     Then,  ^^  =  tan  ^  =  e.     With 

F  as  a  center  and  a  radius  equal  to  HK,  describe  an  arc  of  a 
circle  cutting  HK  in  P  and  P',  The  points  P  and  P'  so  ob- 
tained are  points  on  the  conic. 

In  this  manner,  as  many  points  as  desired  may  be  obtained, 
and  the  conic  sketched  by  drawing  a  smooth  curve  through 
them. 

Evidently,  they  lie  in  pairs  which  are  symmetrical  with  FB 
as  an  axis  of  symmetry.  This  line  FB  is  called  the  axis  of 
the  conic. 

166.  Vertices  of  a  conic.  The  points  of  the  conic  which  lie 
on  the  line  through  the  focus  perpendicular  to  the  directrix 
are  called  the  vertices  of  the  conic. 

To  obtain  these  points,  draw  lines  through  F  inclined  45° 
and  135°  to  the  line  BF.  From  the  points  of  intersection  of 
these  lines  with  BL  drop  perpendiculars  to  BF.  The  feet  of 
these  perpendiculars  are  the  vertices,  as  the  student  can  easily 
show. 

If  e  =  1,  there  is  only  one  vertex,  but  if  e  ^it  1,  there  are  two 
vertices. 

The  figures  on  the  following  pages  show  conies  constructed 
f or  e  =  I,  e  =  1 ,  and  e  =  f . 

EXERCISE  XXXVI 

1.  Plot  in  different  figures  the  conies  for  e  =  |,  e  =  1,  e  =  ^. 

2.  Plot  in  the  same  figure,  using  the  same  directrix  and  focus  for  all 
the  curves,  the  conies  for  e  =  .9,  e  =  1,  e  =  1.1. 

3.  Assume  a  unit  of  distance,  and  taking  the  distance  from  focus  to 
directrix  to  be  1,  2,  .4,  20,  respectively,  construct  the  conies  for  e  =  1. 

4.  Same  as  example  3  for  e  =  f . 

5.  Same  as  example  3  for  e  =  |. 

6.  Prove  that  the  conic  is  tangent  to  the  line  BL  at  the  intersection  of 
BL  and  a  line  through  F  parallel  to  the  directrix. 


/ 

\                                                ^^^ 

^^^                                                                            / 

\                         z 

^                                                                 / 

V                                                         / 

\                              1^ 

A-                         / 

.V               / 

or             / 

! 

\                   '-^ 

Jl 

/ 

_j\ 

^ 

\ 

/ 

V     ^ 

y 

■\ 

/ 

!j 

/ 

"     / 

So 

\  ' 

V 

x] 

^/ 

1 

^  \ 

x/ 

^  / 

\y 

i^ 

^ 

\ 

V 

\ 

."-     / 

\ 

'    J 

\^ 

^ 

194 


FiQ.  137  c. 
196 


196  ANALYTIC  GEOMETRY 

167.  Classification  of  conies.  From  the  constructions  already 
made,  it  is  evident  that  the  general  shape  of  the  conic  depends 
upon  the  value  of  e,  and  that  the  conies  may  be  divided  into 
three  classes,  according  as  e  <  1,  e  =  1,  or  e  >  1. 

A  conic  whose  eccentricity  is  less  than  1  is  an  ellipse ;  one 

of  eccentricity  equal  to  1,  a  parabola; 

^  ■     and  one  of  eccentricity  greater  than  1, 

u\ pP  an  hyperbola.     (See  footnote,  Art.  171.) 


168.    The  equation  of  the  conic  in  rec- 
tangular coordinates.     Let  the  directrix 
„    be  taken  as  ?/-axis  and  the  line  through 
"^fTTo)  t^^  focus  perpendicular  to  the  directrix 

as  the  fl?-axis.     Let  the  distance  from  the 
directrix  to  focus  be  p.     Then  the  coor- 
dinates of  F  are  {p,  0).     Let  P{xy  y)  be 
any  point  on  the  conic,  and  MP  the  distance  from  P  to  the 
directrix.     Then,  from  the  definition  of  the  conic, 

^^^  =  e,  or  FP=eMP. 
MP       ' 

But  FP=:V{x-py  +  y%  and  MP=x. 

.  •.     (x—py-\-y^  =  e^y?^ 

or  (1  -  e2)a?2  _  2  j^a?  +  1,2  4.^2  ^  0. 

This  is,  therefore,  the  equation  of  any  conic  when  the  2/-axis 
is  the  directrix  and  the  a^axis  is  the  line  through  the  focus 
perpendicular  to  the  directrix. 

169.  The  parabola,  e  =  1.  In  the  equation  just  found  let 
e  =  1.     The  conic  is  then  a  parabola.     The  equation  reduces  to 

y^  =  2px—p^. 
This  equation  of  the  parabola  was  obtained  in  Art.  75,  and 
from  the  same  definition  as  here  used.     The  equation  was  dis- 
cussed in  that  place.     The  student  should  review  Arts.  75-78 
at  this  time. 


THE  CONIC  SECTIONS  197 

170.   The  centric  conies.    e=^l.    In  the  equation  of  Art.  168, 

divide  by  the  coefficient  of  a^  and  then  complete  the  square  in 
the  terms  containing  x, 

x^  -    ^P   X  +      ^'      4-     y'     -      -P'      _    P^    _     -P'^' 


1  _  e2         '     (^1  _  g2)2    '     1  _  e2         (^l_  g2^2         1  _  g2         (J^  _  g2^2' 


or 


f       p    \^  I    y^        P'^ 


Substitute  x'  =  x  —  --^—,  y'  =  y, 

1  —  e- 

which    transforms    to    parallel    axes    through    (     ^     .  0 ) 
(Art.  52.)     The  equation  then  becomes 
x"  + 


_       p^€^ 


1-e^     (1-ey 

Dividing  by  the  right-hand  member  brings  the  equation  into 
the  form 

x'^  ^'2 

(1  _  e2)2         1  _  g2 

Since  this  equation  contains  only  even  powers  of  x  and  y, 
the  curve  is  symmetric  with  respect  to  both  coordinate  axes, 
and  hence  with  respect  to  the  origin.  The  origin  may  there- 
fore be  called  the  center  of  the  conic,  and  the  conic  called  a 
centric  conic. 

Also,  since  the  conic  is  symmetric  with  respect  to  the  center, 
rotation  of  the  conic  in  its  own  plane  through  180°  about  its 
center  will  bring  the  conic  back  into  its  original  position,  hav- 
ing merely  interchanged  the  points.  Let  the  conic,  together 
with  its  focus  and  directrix,  be  thus  rotated.  The  focus  and 
directrix  are  brought  into  new  positions  which  are  symmetric 
with  respect  to  the  center.  They  have  remained  focus  and 
directrix  of  the  conic,  however,  and  since  the  new  position  is 


198  ANALYTIC  GEOMETRY 

the  same  as  the  old  position  they  must  be  focus  and  directrix 
of  the  conic  in  its  original  position. 

Therefore  every  centric  conic  has  two  foci  and  two  direc- 
trices. 

They  are  respectively  symmetric  with  respect  to  the  center. 

171.  The  ellipse.  e<l.  In  eq.  (A)  of  the  preceding  article, 
the  divisors  of  x'^  and  2/'^  are  both  positive  if  e  <  1.  For  con- 
venience let 

Substituting  these  values  in  eq.  (A)  and  dropping  primes,  it 
becomes 

This  is  known  as  the  standard  form  of  the  equation  of  the 
ellipse.* 

172.  Axes  of  the  ellipse.  Letting  y  =  0,  the  intercepts  of 
the  ellipse  on  the  a^axis  are  found  to  be  a  and  —  a.  The 
intercepts  on  the  ?/-axis  are  b  and  —  b. 

The  length  2  a  is  called  the  major  axis,  and  2  b  the  minor 
axis. 

The  relation  connecting  a,  b,  and  e  is  found  from  eq.  (1)  of 
the  preceding  article  to  be 

a\l  -  e2)  =  b^ 
This  equation  shows  that  a>b. 

*  In  Art.  83  the  ellipse  was  defined  in  an  altogether  different  way.  The 
equation  of  the  ellipse  derived  from  that  definition  and  that  just  derived 
are,  however,  the  same,  which  proves  that  the  two  definitions  are  equiva- 
lent. The  property  of  the  ellipse  used  in  Art.  83  as  a  definition  will  be 
shown  in  a  succeeding  article  to  follow  from  the  definition  used  in  thia 
chapter. 

A  hke  remark  applies  to  the  hyperbola. 


THE  CONIC  SECTIONS 
Y 


199 


Fig.  139. 
The  abscissa  of  the  new  origin  referred  to  the  old  in  the 


transformations  of  Art.  170  is  — ^ 

1  — e^ 


I.e. 


Now 


a  = 


pe 


l-e^ 


A  BO  = 


e 


Also       FO  =  BO-BF=:^ 


or 


FO  =  ae. 

The   relation  a\l-e^)  =  b^  may  be   written  a^e^  =  a^-^b^f 
from  which 

ae=  y/aP'  -  &2. 
Therefore  if  the  end  of  the  minor  axis  be  taken  as  a  center 
and  an  arc  described  with  the  semi-major  axis  as  a  radius,  this 
arc  will  cut  the  major  axis  in  the  focus. 

173.   Summary.      In   an  ellipse  whose   major   axis   is   2  a, 
minor  axis  2  6,  and  eccentricity  e,  the  following  relations  hold : 


a-^C* 


a^ 


62. 


ae  =  distance  from  center  to  focus, 

—  =  distance  from  center  to  directrix. 
e 


200 


ANALYTIC  GEOMETRY 


174.    The  hyperbola,    e  >  1.    In  eq.  (A),  Art.  170,  the  divisor 
of  ^'^  is  negative  if  e  >  1.     Let  then 


a"  = 


.  ft^ 


e^~l 


Then  both  a  and  h  are  real. 

Substituting  these  values  in  eq.  (A)  and  dropping  primes,  it 
becomes 

^  -  ^  =  1. 


This  is  known  as  the  standard  form  of  the  equation  of  the 
hyperbola. 

(See  also  Art.  87,  and  the  footnote  to  Art.  171.) 

175.   Axes  of  the  hyperbola.     Letting  y  =  0,  the  intercepts 
on  the  a^axis  are  seen  to  be  a  and  —  a.    If  a?  =  0, 2/  is  imaginary. 


Fig.  140. 


Hence  the  curve  does  not  cross  the  y-axis. 
The  length  2  a  is  called  the  transverse  axis,  and  2  b  the  con- 
jugate axis. 


THE  CONIC  SECTIONS  201 

The  relation  connecting  a,  b   and  e   is  6^  =  a^  (e^  —  1),  or 
a^e^  =  a'  +  b\ 

This  shows  that  6  =  a  according  as  e  =  V2. 
As  in  the  ellipse,  the  abscissa  of  the  center  referred  to  the 
d  origin,  ( 
since  e  >  1. 


old  origin,  on  the  directrix,  is  ^  ^    ,,  which  is  here  negative, 

1  —  e'^ 


1  —  e^ 

Now  a  =    .,      ^. 

e^  —  1 

.-.     OB  =  -. 

e 

Also  OF=OB+p 

e^  —  1 

—    P^^ 
~  e^  -  1 

=  ae. 


Since  ae  =  Va^  +  6^,  the  focus  may  be  obtained  by  using  the 
center  of  the  conic  as  a  center  and  the  hypotenuse  of  the  right 
triangle  whose  sides  are  a  and  6  as  a  radius  and  describing  an 
arc  to  cut  the  major  axis  produced. 

176.  Summary.  In  an  hyperbola  of  transverse  axis  2  a,  con- 
jugate axis  26  and  eccentricity  e,  the  following  relations  hold: 

ae  =  distance  from  center  to  focus, 

-  =  distance  from  center  to  directrix. 

e 

Compare  Art.  173. 


202 


ANALYTIC   GEOMETRY 


EXERCISE  XXXVII 

1.  Derive  the  equation  of  the  parabola  whose  directrix  is  the  line  aj  =  6, 
and  whose  focus  is  (2,  3). 

2.  Derive  the  equation  of  an  ellipse  whose  directrix  is  the  line  ?/  =  4, 
focus  at  (0,  2),  and  center  at  (0,  —  1). 

3.  Derive  the  equation  of  the  hyperbola  of  eccentricity  2,  with  focus 
at  (0,  4)  and  the  line  ic  =  2  as  directrix. 

4.  What  is  the  eccentricity  of  the  equilateral  hyperbola  ? 

5.  Keeping  the  major  axis  unchanged,plot  ellipses  with  eccentricity  .1, 
.5,  .9. 

What  limiting  position  do  the  foci  approach  as  the  eccentricity  ap- 
proaches the  limit  0  ?     What  is  the  limiting  form  of  the  ellipse  ? 


177.   The  equation  of  the  conic  in  polar  coordinates. 

(a)  Origin  at  the  focus.  Taking  the  origin  at  the  focus  and 
the  initial  line  perpendicular  to  the  di- 
rectrix, the  polar  equation  of  the  conic 
is  easily  written. 

Let  P(r,  6)  be  any  point  on  the  conic 
and  MP  the  length  of  the  perpendicu- 
lar from  P  to  the  directrix.  Then,  by 
the  definition  of  the  conic, 

FP=eMP, 

or  r  =  e(p  -f  r  cos  $), 

from  which 


Fig.  141. 


ep 


1  —  c  cos  8 

If  the  focus  lies  to  the  left  of  the 
directrix,  then 

PM=p  —  r  cos  $. 
.'.     r  =  e(p  —  r  cos  6)j 
from  which 

^- ep 


1  4-  c  cos  6 


Fig.  142. 


THE  CONIC  SECTIONS  203 

(b)  Origin  at  the  center.     For  the  centric  conies  the  equa- 
tion  in  rectangular  coordinates  is 

the   upper   sign    being   for    the   ellipse,   the    lower    for  the 
hyperbola. 

Change  to  polar  coordinates  by  means  of 

x  =  r  cos  6, 
y  =  r  sin  6. 

Substituting  and  clearing  of  fractions, 

T^b^  cos^^  ±  ?%2  sin2^  =  a^b% 
from  which 

a'b' 


r2  = 


62  cos^^  ±  a'  sin^^ 


This  equation  may  be  expressed  in  a  somewhat  simpler 
form  in  terms  of  the  eccentricity  and  6.  For  convenience  con- 
sider separately  the  equation  of  the  ellipse.     It  is 


or 


since 


62cos2^  +  a2sin2^     b'^co^'^O  +  a 

2(1  -  C0S2^) 

62 

a' 

C0S2^, 

^._        h'^ 

l-e^cos^e' 

e^-(«^-^r 

Similarly  for  the  hyperbola  the  equation  is 


204  ANALYTIC  GEOMETRY 

EXERCISE  XXXVIII 
Determine  the  nature  of  the  following  conies  and  sketch  them: 
1.  r  = 

3.   r  = 

5.   r^  = 

2+3  sin2d 

7.   r^  = ""  8.   r  =  a  sec2.^ 

16-20sin2(9  2 

9.   Show  that  if  the  vertex  of  a  parabola  is  taken  as  origin  and  the 
axis  of  the  parabola  as  the  initial  line,  the  equation  in  polar  coordinates  is 

2  p  cos  d 

r  =    ^.   ^ 

sni2^ 


4 

1- 

- 1  cos  e 

5 

2- 

-  2  cos  0 

3 

2 

-  C0S2  d 

64 

2 

r  = 

3 

2  +  4  COS  ^ 

4. 

r'2 

-6 

1  -  4  cos20 

a 

^2 

20 

CHAPTER   XIV 

PROPERTIES  OF  CONICS 

178.     In  this  chapter  a  few  of  the  more  important  properties 
of  the  conies  are  derived. 


I.  PROPERTIES  OF  THE  PARABOLA 

179.  Subtangent  of  the  parabola.     In  Art.  130  the  equation 
of  the  tangent  to  the  parabola  7f==2px  at  (iCo,  y^  was  found  to  be 

Letting  ?/  =  0  in  this  equation,  there  results  x  =  —  Xq, 
i.e.         0T=  -a-oCFig.  143). 
.-.       TO  =  Xq. 
.'.      TM==2xo. 

The  line  TM  is  called  the 
subtangent. 

180.  The  subnormal  of  the 
parabola.  The  slope  of  the  nor- 
mal to  the  parabola  y^  =  '2px 
at  the  point  {xq,  y^)  is  the 
negative  reciprocal  of  the  slope 
of  the  tangent  at  that  point; 

i.e.  the  slope  of  the  normal   is 


?h 


P 


Fig.  143. 
The  equation  of   the 


normal  is  therefore 


^0, 


205 


206 


ANALYTIC  GEOMETRY 


To  find   where  the  normal  cuts  the  a;-axis,  let  y  =  0.     The 
result  is 

I.e.  in  Fig.  143.  OiV=  o^o  +jp. 

MN=p. 

The  line  MN  is  called  the  subnormal. 

Hence,  in  the  parabola  the  suhnormal  is  constant  and  equal  to  p. 

181.     Property  of  reflection  of  the  parabola.     In  Fig.  143, 
Art.  179,  from  the  definition  of  the  parabola, 

z 

Also  TF=  TO-\-OF  =  x,  +|.    (Art.  179.) 

FP=TF. 
Z.FPT  =  ^FTP=Z.TPH. 

Let  PL  be  drawn  parallel  to  the  axis  of  the  parabola.     Then 

AFPT=ZLPQ. 

Hence,  if  the  parabola  were  a  reflector,  any  ray  of  light  from 

the  focus  striking  the  parabola  and  reflected  so  as  to  make  the 

angle  of  reflection  equal  to  the  angle 

of  incidence  would  be  reflected  along 

a  parallel  to  the  axis  of  the  parabola. 

A  concave  reflecting  surface  in  the 
form  of  a  surface  generated  by  re- 
volving a  parabola  about  its  axis 
would  therefore  reflect  all  rays  from 
a  source  at  the  focus  in  lines  parallel 
to  the  axis  of  the  reflector. 

Definition.  The  chord  of  a  conic 
which  passes  through  the  focus,  per- 
pendicular to  the  axis  of  the  conic,  is 
Fig.  144.  called  the  latus  rectum. 


PROPERTIES  OF  CONICS  207 


EXERCISE  XXXIX 

1.  By  means  of  the  result  found  in  Art.  179,  show  how  to  draw  a  tan- 
gent at  any  point  of  the  parabola. 

2.  Prove  that  the  tangents  at  the  ends  of  the  latus  rectum  meet  at  the 
intersection  of  the  directrix  and  the  axis  of  the  parabola,  and  are  at  right 
angles  to  each  other. 

3.  Prove  that  the  distance  from  the  focus  of  a  parabola  to  a  tangent  is 
half  the  length  of  the  normal  from  the  point  of  tangency  to  the  axis  of  the 
parabola. 

4.  Prove  that  any  point  P  of  the  parabola  and  the  intersections  of  the 
axis  of  the  parabola  with  tangent  and  normal  at  P  are  all  equidistant 
from  the  focus. 

5.  Prove  that  the  tangent  at  any  point  of  a  parabola  meets  the  directrix 
and  latus  rectum  produced  at  points  equally  distant  from  the  focus. 

6.  Show  that  the  normal  at  one  extremity  of  the  latus  rectum  of  a 
parabola  and  the  tangent  at  the  other  extremity  are  parallel. 

7.  Show  that  the  directrix  of  a  parabola  is  tangent  to  the  circle  described 
on  any  chord  through  the  focus  as  a  diameter. 

8.  Show  that  the  tangent  at  the  vertex  of  a  parabola  is  tangent  to  the 
circle  described  on  any  focal  radius  as  a  diameter. 

9.  Prove  that  the  angle  between  two  tangents  to  a  parabola  is  equal  to 
one  half  the  angle  between  the  focal  chords  drawn  to  the  points  of  contact. 

10.  Prove  that  the  tangents  at  the  ends  of  any  focal  chord  of  a  parabola 
meet  on  the  directrix. 

11.  Prove  that  the  length  of  the  latus  reetum  of  the  parabola  y^  —  Ipx 
is  2  p. 

12.  Prove  that  if  from  a  point  (a;o,  2/o)  two  tangents  are  drawn  to  the 
parabola,  the  equation  of  the  line  through  the  points  of  tangency  is 
2/oy  =p{x  +Xo). 

13.  By  means  of  the  preceding  example  prove  that  if  tangents  are  drawn 
to  the  parabola  from  any  point  on  the  directrix,  the  line  through  the  points 
of  tangency  passes  through  the  focus. 

14.  Prove  that  in  the  parabola  if-  =  2  pa;,  the  ordinate  of  the  middle 

point  of  a  chord  of  slope  m  is  — ,  and  hence  that  the  locus  of  the  middle 

m 

P 
points  of  a  system  of  parallel  chords  of  slope  m  is  the  straight  line  y  =  — 


m 


Draw  the  figure. 


208  ANALYTIC  GEOMETRY 

Definition.  The  straight  line  which  bisects  a  system  of  parallel 
chords  of  a  parabola  is  called  a  diameter  of  the  parabola. 

15.  Find  the  equation  of  the  diameter  which  bisects  all  chords  of  slope 
m  in  the  parabola  x^  =  2py.     Ans.  x  =  mp. 

16.  Transform  the  equation  of  the  parabola  y^  =  2  px  to  the  tangents 
at  the  extremities  of  the  latus  rectum  as  axes. 

Suggestion.     First,  moving  to  parallel  axes  through   (  — ^,    0),  the 

equation  becomes 

y^  =  2px-  p^. 

Next,  rotating  the  axes  through  —  45°^  the  equation  becomes 
a;2 -2xy-\-y^-  2V2p(x  +  y)  +  2p^  =  0, 

which  becomes  a  perfect  square  on  the  left  by  the  addition  of  4  xy. 

Then  extract  square  root,  transpose,  extract  square  root  again,  and 
obtain 

Vx±Vy  =  ±  yp\/2, 
or  Vx  ±Vy  =  ±  Va, 

where  a=pV2. 

17.  Plot  the  curve 

x^±y^  =  ±  a\ 
"What  portions  of  the  curve  correspond  to  the  different  combination  of 
signs  ? 

II.  PROPERTIES  OF  THE  ELLIPSE  AND  OF  THE 
HYPERBOLA 

182.  Focal  radii  of  the  ellipse.  Let  P(xo,  y^  be  any  point  of 
the  ellipse  of  semi-axes  a  and  6,  and  let  r  and  r'  be  the  radii 
from  the  foci  F  and  F^  to  P. 

Through  P  draw  a  line  parallel  to  the  major  axis  of  the 
ellipse,  meeting  the  directrices  in  M  and  M\  Then  from  the 
definition  of  the  ellipse,  using  the  left-hand  focus  and  directrix, 


M'P 


or  r^  =^eM'P=e[--\-XQ\  =  a  +  exQ. 


PROPERTIES  OF  CONICS 


209 


Similarly,  using  the  right-hand  focus  and  directrix, 

r  =  ePM=ef-  —  XQ\=a  —  exo. 
Adding,  r-{-r'  =  2a. 

Hence  the  sum  of  the  focal  radii  of  a  point  on  the  ellipse  is 
constant,  and  equal  to  the  major  axis  of  the  ellipse. 


Fig.  145. 

183.  Focal  radii  of  the  hyperbola.  In  a  manner  similar  to 
the  above  the  student  can  show  that  in  the  hyperbola  the  focal 
radii  are  r  =  exo  -\-  a  and  r'  =  exo  —  a,  and  hence 

r  —  r'  =  2  a. 

184.  Property  of  reflection  of  the  ellipse.  The  focal  radii  to 
any  point  of  an  ellipse  make  equal  angles  with  the  normal  to 
the  ellipse  at  that  point. 

Proof.    In  Art.  130,  the  equation  of  the  tangent  to  the  ellipse 

at  (xq,  2/o)  was  found  to  be 


yvo 


1. 


The  slope  of  the  normal  at  (xq,  y^)  is  therefore  J^,  and  the 

b-XQ 


210  ANALYTIC  GEOMETRY 

equation  of  the  normal  is 

2/-2/o  =  ||(«'-^o). 

In  this  equation  let  y  =  0  and  solve  for  x, 

x  =  Xo f  = -—  Xq  =  e'xQ. 

I.e.  in  Fig.  145. 

0N=  e%. 

F'N=  ae  +  e^XQ  =  e(a-\-  ex^), 
an  d  NF  =  ae  —  s^Xq  =  e  (a  —  gxq)  . 

gJ^=^  +  ^^o  =  ^(Art.l82). 
NF     a  —  exQT 

Therefore  by  plane  geometry,  Z  F'PN  =  Z  NPF,  which 
proves  the  theorem.  Hence  if  the  ellipse  served  as  a  reflector, 
a  ray  of  light,  or  sound,  emitted  at  one  focus  would  be  reflected 
to  the  other. 

It  is  on  this  principle  that  whispering  galleries  are  some- 
times constructed. 

185.  Property  of  reflection  of  the  hyperbola.  In  the  hyper- 
bola the  focal  radii  to  any  point  of  the  curve  make  equal 
angles  with  the  tangent  at  that  point. 

The  proof  is  left  to  the  student. 

186.  If  a  line  is  drawn  to  cut  the  hyperbola  in  two  points, 
the  two  segments  of  the  line  included  between  the  hyperbola 
and  its  asymptotes  are  equal. 

Proof.  The  equations  of  the  hyperbola,  its  asymptotes,  and 
any  line  are,  respectively, 

b^x'-ay^a^  (1) 

b''x'-aY=0,  (2) 

and  y=mx-\-c.  (3) 


PROPERTIES  OF  CONICS 


211 


Let  the  points  of  intersection  of  line  and  hyperbola  be 
A(^ij  2/1)  ^^^  A(^2)  2/2);  and  of  line  and  asymptotes  be  Qi(xi',  y^) 
and  Q.lx.J,  yJ). 

Substituting  the  value  of  y  from  eq.  (3)  in  eqs.  (1)  and  (2) 
respectively,  and  collecting  terms,  there  results 

(62  _  a'nv^x'  -  2  a^mcx  -  a'ic'  +  W)  =  0  (4) 

and  (6  —  a?m^)x^  —  2  ahncx  —  oj^c?  =  0.  (5) 

The  roots  of  (4)  are  x^  and  x^,  and  of  (5)  are  x-^  and  x^. 
Now  in  any  quadratic  equation  the  sum  of  the  roots  is  equal 
to  minus  the  coefficient  of  the  first  power  of  the  variable 
divided  by  the  coefficient  of  the  second  power ;  and  since  the 
first  two  terms  in  eqs.  (4)  and  (5)  are  the  same,  therefore 

Ju\  ~p"  Ct/o  —  *vi      -y"  iCo  • 
/y»        I      rp  rp  '      |      rp   ' 

But  zLX_2  and  ^  "^  ^  are  respectively  the  abscissas  of  the 
middle  points  of  P1P2  and  Q1Q2. 


Fig.  146. 


the  middle  points  of  P1P2  and  Q1Q2  coincide. 


Q.E.P. 


212  ANALYTIC  GEOMETRY 


EXERCISE  XL 

1.  Prove  that  the  length  of  the  latus  rectum  of  an  ellipse  or  an  hyper- 
bola is  . 

a 

2.  Prove  that  the  tangents  at  the  extremities  of  the  latus  rectum  of 
an  ellipse  or  hyperbola  intersect  on  the  directrix. 

3.  Prove  that  the  line  drawn  from  the  focu^  to  the  intersection  of  a 
tangent  and  the  directrix  of  an  ellipse  or  hyperbola  is  perpendicular  to 
the  line  from  the  focus  to  the  point  of  tangency. 

4.  A  circle  is  drawn  on  the  major  axis  of  an  ellipse  as  a  diameter. 
A  perpendicular  to  the  major  axis  meets  the  ellipse  and  circle  in  P  and 
Q  respectively.  Prove  that  the  tangents  drawn  at  F  and  Q  intersect  on 
the  major  axis.  Hence  show  how  to  construct  a  tangent  to  an  ellipse  at 
a  given  point. 

5.  Show  that'  the  distance  from  the  focus  to  an  asymptote  of  an 
hyperbola  is  equal  to  b. 

6.  Prove  that  the  product  of  the  perpendiculars  from  any  point  of  an 
hyperbola  upon  the  asymptotes  is  constant,  and  equal  to  —  • 

7.  Prove  that  the  product  of  the  perpendiculars  from  the  foci  upon  a 
tangent  to  the  ellipse  is  equal  to  the  square  of  the  semi-minor  axis. 

8.  State  and  prove  a  like  property  of  the  hyperbola. 

JW.2        oi2 

9.  Prove  that  if  tangents  are  drawn  to  the  ellipse  —  +  f-  =  1  from  an 
exterior  point  (Xq,  ?/o),  the  equation  of  the  line  through  the  points  of  tan- 
gency is  ^  +  ^  =  1 . 

10.  Prove  the  statement  in  example  9  to  be  true  for  the  hyperbola, 
with  proper  changes  of  sign. 

11.  Prove  that  if  tangents  are  drawn  to  an  ellipse  or  hyperbola  from 
any  point  on  the  directrix,  the  line  joining  the  points  of  tangency  passes 
through  the  focus.     (Use  examples  9  and  10.) 

12.  Through  a  fixed  point  within  a  given  circle,  a  circle  is  drawn  tan- 
gent to  the  given  circle ;  prove  that  the  locus  of  its  center  is  an  ellipse. 
Draw  the  figure. 


PROPERTIES  OF  CONICS  213 

13.   A  line  y  =  mx  ■}-  c  cuts  the  ellipse  b^x^  +  a^y^  =  a^b^ ;  prove  that 
if  (xi,  yi)  is  the  middle  point  of  the  chord,  then 
a^mc         ,.  b'^c 


b'^-\-a^m-^  b^  +  a^m^ 

14.  From  the  preceding  example,  by  eliminating  c,  prove   that   the 

locus  of  the  middle  points  of  a  system  of  parallel  chords,  with  slope  m,  of 

the  ellipse  is  the  straight  line 

&2 
y  = X. 

This  line  is  called  a  diameter  of  the  ellipse. 

Prove  that  any  line  through  the  center  of  an  ellipse  is  a  diameter. 

15.  Show  that  if  two  lines  through  the  center  of  the  ellipse 

6^x2  +  a2y2  =  ^252 

have  slopes  m  and  m'  such  that  mm'  = ,  then  each  line  bisects  all 

chords  parallel  to  the  other. 
Draw  two  such  lines. 
Two  such  lines  are  called  conjugate  diameters. 

/y2  o<2 

16.  Prove  that  in  the  hyperbola ^  =  1  the  equation  of  the  locus 

of  the  middle  points  of  a  system  of  parallel  chords  of  slope  m  is 

y= — zx. 

17.  Through  the  point  (xq,  yo)  on  the  ellipse  b^x^  +  a^y^  =  a262  a 
diameter  is  drawn  ;  prove  that  the  coordinates  of  the  extremities  of  its 

conjugate  diameter  are  a:  =  ±  ^^,    y  =  T  — . 

b  a 

18.  If  a'  and  6'  are  the  lengths  of  two  conjugate  semi-diameters  of 
the  ellipse,  prove  that  a'^  -\-  b''^  =  a^  +  b^.     (Use  example  17.) 

19.  Prove  that  the  tangent  at  any  point  of  the  ellipse  is  parallel  to  the 
diameter  which  is  conjugate  to  the  diameter  through  the  given  point ; 
and  hence  that  the  tangents  at  the  extremities  of  two  conjugate  diameters 
form  a  parallelogram. 

20.  Prove  that  the  area  of  the  parallelogram  formed  by  the  tangents 
at  the  extremities  of  two  conjugate  diameters  of  an  ellipse  is  constant, 
and  is  equal  to  4  ab. 

Suggestion.  The  area  in  question  is  8  times  the  area  of  the  triangle 
whose  vertices  are  (0,  0),  (xo,  yo),  and  (^   -  ^\ .     (See  example  17.) 


CHAPTER   XV 

THE   GENERAL   EQUATION  OF   SECOND  DEGREE   IN  TWO 
VARIABLES 

187.  In  the  preceding  chapters  certain  equations  of  second 
degree  in  two  variables  have  been  studied.  It  will  now  be 
shown  that  every  equation  of  second  degree  in  two  variables 
with  real  coefficients  is  the  equation  either  of  one  of  the  conies, 
a  circle,  a  pair  of  straight  lines,  one  straight  line,  a  point,  or 
else  the  equation  has  no  locus. 

Moreover,  the  conditions  which  the  coefficients  must  satisfy 
in  the  different  cases  will  be  established. 

188.  The  general  equation  of  second  degree  in  x  and  y  is 

aa?2  +  hocy  -\^  cij^  +  dijc-\-  ey  +  f  =  0.  (1) 

Let  the  origin  be  moved  by  a  translation  of  axes  to  the  point 
(h,  k)  by  means  of  the  formulas 

x  =  x'  -{-hj 
y  =y'  -{-k. 

Equation  (1)  then  becomes 

ax'^  +  bx'y'  +  cy"  +  d'x'  +  e'y'  +/'  =  0,  (2) 

where                   d' =  2  aJi  +  bk  +  d,  (3) 

e'  =  6/i  +  2  cA;  -f  e,  (4) 

/  =  ah^  +  bhk  -f  ck^  +  dh  +  ek  -\-f.  (5) 

Equation  (2)  will  be  simplified  if  h  and  k  can  be  so  chosen 
that  d'  =  0  and  e'  =  0.  Putting  d'  =  0  and  e'  =  0  and  solving 
for  h  and  k, 

J,  _2cd—7}e      7.  _  2  ae  -  bd  /n\ 

214 


THE  GENERAL  EQUATION  OF  SECOND  DEGREE   215 

These  values  of  li  and  k  are  definite  finite  values  unless 
6^  —  4  ac  =  0,  in  which  cases  there  are  no  values  of  h  and  h 
that  make  d'  =  0  and  e'  =  0. 

Hence  there  are  two  cases  to  consider,  I,  6^  —  4  ac  ^  0,  and 
II,  h^-4.ac  =  0. 

Case  I.     h^-^ac^(i, 

189.  Removal  of  the  terms  of  first  degree.  Consider  first 
the  case  where  b^  —  ^ac^O.  Then  if  Ti  and  k  have  the  values 
shown  in  eq.  (6),  d'  and  e'  are  both  zero,  and  eq.  (2)  becomes 

a«'2  +  6xy  +  c?/'2  +/  =  0.  (7) 

The  value  of  /'  can  be  obtained  by  substituting  the  values  of 
h  and  k  from  (6)  in  (5),  but  more  easily  as  follows :  Multiply 
eq.  (3)  by  h,  eq.  (4)  by  k,  and  add.     The  result  is 

dli  +  e'A:  =  2  ah''  +  2  hhk  -\-2ck'  +  dh  +  ek. 
To  both  members  of  this  equation  add  dh  -\-  ek  -\-  2/.     Then 
d'h  +  e'k  +  dh  +  eA:  +  2f=:2{ah^  -f  bhk  +  cF  +  d/i  +  ek-\-f)=2f 
or  2f'  =  dh-\-ek-\-2f,    since  d' =  e' =  0. 

Substituting  the  values  of  h  and  k  from  eq.  (6), 

f  _  —  (4  acf-]-  bde  —  ae^  —  cd^  —fb^  /on 

The  quantity  in  the  parenthesis  is  of  importance  in  what  fol- 
lows.    For  convenience  let  it  be  denoted  by  a  single  letter,  H; 

H=  4  acf  +  bde  -  ae^  -  cd^  -  fh\ 
Also  let  I>  =  h^-^.ac. 

190.  Removal  of  the  term  in  ocy.  Equation  (7)  may  be  re- 
duced to  one  lacking  the  a^y-term  by  a  proper  rotation  of  the 
axes. 

Let  a;'  =  oj"  cos  6  —  ly"  sin  ^, 

2/'  =  a?"  sin  0  -f  y"  cos  0. 
Substituting  these  values  in  eq.  (7),  it  becomes 

a!x''''  +  b^x'Y  +  cy"  +  /'  =  0,  (9) 


216  ANALYTIC  GEOMETRY 

where 

a'  =  a  cos^  6  -\-b  cos  ^  sin  ^  +  c  sin^  6^  (10) 

6'  =  -  2  a  cos  ^  sin  ^  -f  6  (cos^  0  -  sin^  ^)  +  2  c  cos  ^  sin  0,     (11) 
c'  =  a  sin2  ^  _  5  sin  ^  cos  ^  +  c  cos^  d.  (12) 

Now  let  0  be  so  chosen  that  b'  =  0,  i.e.  let 

b  (cos^  $  —  sin^  ^)  =  2  (a  —  c)  cos  6  sin  ^, 
or  &  cos  2  ^  =  (a  —  c)  sin  2  ^, 

or  tan  2  0  -  — ^.  (13) 

tt  —  c 

Since  the  tangent  of  an  angle  may  have  any  real  value,  it  is 
always  possible  to  choose  0  so  that  b'  =  0. 
With  this  value  of  6j  eq.  (9)  becomes 

a'i€"'  +  c'y"'^+f'  =  0.  (14) 

191.  Locus  of  the  equation.  The  nature  of  the  locus  of  eq. 
(14)  depends  upon  the  signs  of  a',  c',  and/',  and  these  signs 
depend  upon  the  original  coefficients  of  eq.  (1). 

To  determine  the  signs  of  a'  and  c'  one  may  proceed  as  fol- 
lows: Using  the  relations 

2  sin  ^  cos  ^  =  sin  2  Oy 

2  cos^  ^  =  1  4-  cos  2  ^, 
2  sin2  ^  =  1  -  cos  2  ^, 

eqs.  (10)  and  (12)  may  be  written 

2a'  =  a  +  c  +  ?>  sin  2^  +  (a  -  c)  cos  2^.  (15) 

2c'  =  a  +  c  -  5  sin  2^  -  (a  -  c)  cos  2^.  (16) 

Adding,  a'  +  c'  =  a -[- c.  (17) 

Subtracting,  a'  —  c'  =  6  sin  2  ^  -f-  (a  —  c)  cos  2  0. 

From  equation  preceding  (13), 

6  cos  2  ^  -  (a  -  c)  sin  2  ^  =  0. 


THE  GENERAL  EQUATION  OF  SECOND  DEGREE  217 

Square  and  add 

&  sin  2  (9  +  (a  —  c)  cos  2  ^  =  a'  —  c', 
and  6  cos  2  ^  —  (a  —  c)  sin  2  ^  =  0, 

and  there  results 

62  +  (a  _  cf  =  (a'  -  c^. 
Erom  (17)  (a  +  cf  =  (a'  +  cj. 

Subtracting,  4  a'd  =  4.ac-  b\  (18) 

Since  4.ac  —  b^  =^0,  neither  a'  nor  c'  can  be  zero.     Eq.  (14) 

may  therefore  be  written,  since  from  eq.  (8),/'  = , 

no  f^2 

^  =  1,      HH^O,  (19) 


//     '     H 


a'D        c'D 

or  a'x"'  +  c'y"'  =  0,      itH=0,  (20) 

Two  cases  must  here  be  considered. 

(1)  Z)<0,  ^.e.  4ac-&2>0. 

Then  neither  a  nor  c  can  be  zero,  and  a  and  c  must  be  of  like 
signs.  It  follows  also  from  eq.  (18)  that  a'  and  c'  must  be  of 
like  signs,  and  hence  of  the  same  sign  as  a  and  c,  by  (17). 

TT 

Therefore,  if  —  <  0  the  locus  of  (19)  is  an  ellipse  if  a'  =^  c', 
and  a  circle  if  a'  =  c'. 

TT 

If  —  >  0,  eq.  (19)  has  no  locus. 

Equation  (20)  is  satisfied  only  by  the  point  x"  =  0,  y"  =  0. 

(2)  2»0,  i.e.  4ac-62<o. 

It  follows  from  (18)  that  a'  and  c'  are  of  opposite  signs. 

Equation  (19)  is  therefore  the  equation  of  an  hyperbola 
whether  H  is  positive  or  negative. 

Equation  (20)  can  be  factored,  and  its  locus  is  therefore  the 
pair  of  intersecting  straight  lines 

^s/¥x"  +  V^^y"  =  0,    -VaJx"  -  ^^7y"  =  0. 


218  ANALYTIC  GEOMETRY 

192.  Condition  that  eq.  (1)  represents  a  circle.  It  was  shown 
in  the  preceding  article  that  the  locus  of  eq.  (1)  is  a  circle  when 

TT 

D  <0,  —  <  0,  and  a'  =  c'.     The  third  of  these  conditions  can 
a 

be  expressed  in  terms  of  the  original  coefficients  of  eq.  (1)  as 

follows :  In  (17)  and  (18)  put  a'  =  c'.     Then 

2  a'  =  a  +  c, 

and  4a'^  =  4ac  — 6^. 

Substituting  in  the  second  of  these  equations  the  value  of  a' 
from  the  first,  there  results 

(a  -  cf  +  6^  =  0. 

This  can  be  satisfied  by  real  values  of  a,  b,  and  c  when  and 
only  when  a  =  c  and  b  =  0. 

Hence  the  conditions  that  eq.  (1)  represents  a  circle  are 

I)<0,  —  <  0,  6  =  0,  and  a  =  c. 
a 

Case  II.     b^-4:ac  =  0. 

193.  Pass  now  to  the  case  where  6^  —  4  ac  =  0.  In  this  case 
not  both  a  and  c  can  be  zero,  for  then  b  would  be  zero  and  eq.  (1) 
would  be  only  of  the  first  degree.  Moreover,  a  and  c  must  be 
of  the  same  sign  if  neither  is  zero.  Assume  at  first  that  sign 
to  be  positive.     Then  eq.  (1)  may  be  written 

asc^  ±  2^acxy  -\-  cy^  +  dx  -\-  ey  -^  f  =  0, 
or  ( Vaa:  ±  -VcyY  -{- dx -\- ey  +  f  =  0,  (21) 

the  ±  sign  being  chosen  according  as  b  is  positive  or  negative. 
In  this  formula  Va  and  Vc  are  real  and  positive. 
Choose  now  an  angle  0  such  that 

Va  =kcos  0,   ±  ^c  =  7c  sin  $,  (22) 

Squaring  and  adding, 

k  =  V^+^,  A;  >  0.  (23) 


THE  GENERAL  EQUATION  OF  SECOND  DEGREE   219 

Then         Vaic  ±  -yjcy  =k(xcos  6  +  y  sin  0). 

Transform  now  to  axes  which  make  the  angle  0  with  the  axes 
X  and  2/j  for  which  the  formulas  of  transformation  are 
X  =  x'  cos  6  —  y'  sin  9, 
y  =  x'  sin  6-{-y'  cos  0. 
Then  x  cos  O  +  y  sin  6  =  x',  and  eq.  (21)  becomes 

kV  +  d'x'  +  e'y'  +/=  0,  (24) 

where  d'  =  d  cos  0  +  e  sin  ^,  (25) 

e'  =  —  dsmO  +  e  cos  ^.  (26) 


If  e'  =^  0  eq.  (24)  may  be  written 
This  is  of  the  form 


yf^-^x"-^x'-l.  (27) 

^  e'  e'         e'  ^    ^ 


y  =  aa^  -j-  5a;  -|-  c 

which  in  Art.  81  was  seen  to  be  the  equation  of  a  parabola. 
If  e'  =  0  eq.  (24)  becomes 

kV  +  d'x'  +/=  0.  (28) 

This  is  a  quadratic  in  x'  alone.     It  is  satisfied  by 


^,_-d'  ±^d"-Ak^f 

X  — • 

2  k' 

Hence  eq.  (28)  is  the  equation  of  two  parallel  lines,  one  line,  ot 
has  no  locus  according  as 

d'^-4:k'f=0. 

< 

194.  Evaluation  of  e'  and  of  d'^  —  4  F/.  The  quantities  e' 
and  d'^  —  4:kyof  the  preceding  article  may  be  expressed  in 
terms  of  the  coefficients  of  eq.  (1)  as  follows :  From  (26)  and  (22) 


.'=-c^(±^^)+ff»  =  l(«VST<iVc). 


(29) 


220  ANALYTIC  GEOMETRY 

Now  since    Z>  =  0 ;  6^  =  4  ac,  and  H  becomes 
H=  bde  —  ae^  —  c(P 

=  ±  2  Vac  •  de  —  ae^  —  cd^ 
=  _(eVaT(^Vc)2, 
or,  from  (29),  il=  -k^e'^.  (30) 

.'.  e'  vanishes  or  does  not  vanish  according  as  H  does  or  does 
not  vanish. 

Again,  from  (25)  and  (22) 

d'  =-(dVa  ±  eVc), 
and  hence         d'^  - 4  A^y  =  \{dVa±  eVcf  -  4  hj 

=  i  iad""  ±  2  de  Vac  +  e^c  -  4/(a  +  c)^]. 

But  if  e'  =  0,  then  eVa  =  ±  ^a/c,  from  eq.  (29). 
/.     d'2  _  4  ;t2y._  1  L^2  4_  2  ^2^  +  !^'  _  4/(a  +  c)2l 

^  (g  +  cf  ^  d'-iaf 
k^  a 

a-\-c 


(d'-4:af). 


Hence  the  sign  of  d'^  —  4  k^f  is  the  same  as  the  sign  of 
d^  —  4:  af.  Therefore  if  /)  =  0  and  a  is  positive,  the  locus  of 
(1)  is  a  parabola  if  H=^0,  and  is  two  parallel  lines,  one  line, 
or  there  is  no  locus  according  as 

d^-4:af=  0,iiH=0. 

< 
If  a  is  negative,  eq.  (1)  may  be  divided  by  —  1  and  then  the 
above  conditions  hold  if  each  coefficient  in  H  and  d^  — A  af  is 
changed  in  sign.  This,  however,  only  changes  the  sign  of  H 
and  does  not  affect  at  all  d^  —  4  af.  The  above  conditions  hold, 
therefore,  whether  a  is  positive  or  negative. 


THE  GENERAL  EQUATION  OF  SECOND  DEGREE   221 

If  a  =  0,  then  6  =  0,  and  c=^0.     Eq.  (1)  then  reduces  to 

c/  +  da;  +  e2/+/=0,  (31) 

or  dx=  —  cif  —  ey  —f. 

This  is  a  parabola  if  d  ^fc  0. 

When  a  =  6  =  0,  H  becomes  —  ccZ^,  and  since  c=^0,  H  van- 
ishes or  does  not  vanish  according  as  d  does  or  does  not  vanish. 
If  H=0  eq.  (31)  becomes 

c/  +  e2/4-/=0, 
which  is  satisfied  by 

„  _  —  e  ±  Ve^  —  4  c/' 
^~  2~c  "' 

the  locus  of  which  is  two  parallel  lines,  one  line,  or  there  is  no 
locus  according  as 

e2_4c/=0. 

< 

195.   Summary.     The  nature  of  the  locus  of  the  general 
equation  of  the  second  degree 

ax^  4-  bocy  +  cy^  -^dx-^ey  +/=  0 
is  shown  in  the  following  table,  in  which 

H=  4  acf+  bde  -  ae^  -  cd^  -fh\ 

(aH  <  0  ellipse,  reducing  to  a  circle  if  6  =  0  ana  a  =  c, 
aH  >  0  no  locus, 
^  =  0  a  point. 

jj^^\    il  9^0  hyperbola, 

I    H  =0  two  intersecting  straight  lines, 
H=^  0  parabola, 

'  a  :^  0  two  parallel  lines,  one  line,  or  no  locus 


D  =  0 


H=0 


according  as  d^—  4  a/  =  0, 
a  =  0  two  parallel  lines,  one  line,  or  no  locus 
according  as  e^  —  4:cf  =  0. 


222  ANALYTIC  GEOMETRY 


EXERCISE  XLI 

Apply  the  above  test  to  determine  the  nature  of  the  loci  of  the  follow- 
ing equations. 

1.  x^-2xy-{-Sy^  +  2x-y  +  3  =  0. 

2.  Sx^-4:xy  +  y^-x  +  2y-l  =  0. 

3.  Sx^  +  6xy-2y^-'Sx-{-y  =  0. 

4.  9x^-6xy  +  y^-Sx-\-y-2  =  0. 

5.  x2-4x?/  +  4  2/2  +  2a:-4?/  +  l  =  0. 

6.  x2-xy  +  y2  +  2a;  +  y  +  2  =  0. 

7.  3x^-Sxy+3y^  +  6x  +  Zy+'7  =  0. 

8.  ^x^-4:xy  +  y^  +  ix-2y  +  2  =  0. 

9.  Ax^  -  12 xy  +  9 y^  +  x - y  +  1  =0. 

10.  3 x^ - xy  -  y^ -\-  X -2  y  -{- 1  =  0. 

11.  Show  that  the  locus  of  2x^  —  2xy  +  y^-Sx  +  y+f=0  is  an 
ellipse,  a  point,  or  there  is  no  locus,  according  as  /  is  less  than,  equal  to, 
or  greater  than  f . 

12.  Show  that  the  locus  of  ax^  +  bxy  -\-cy^  =  0  is  two  intersecting 
lines,  one  line,  or  a  point,  according  as  6^  _  4  ac  is  greater  than,  equal  to, 
or  less  than  zero. 

13.  Show  that  the  locus  of  icy  +  dx  +  ej/  +  /  =  0  is  an  hyperbola  except 
when  /  =  de.     What  is  the  locus  then  ? 

14.  In  the  equation  {Ix  -\-my  +  ny  +  px  +  qy-\-r  =  Q  show  that  2)  =  0, 
and  that  H=  —  {mp  —  IqY,  and  hence  that  the  locus  of  the  equation  is  a 

parabola  except  when  -  =  — .     What  is  the  locus  then  ? 


CHAPTER   XVI 

EMPIRICAL   EQUATIONS 

196.  Statement  of  the  problem.  It  is  sometimes  desirable  to 
find  an  equation  of  a  curve  drawn  through  points  determined 
by  pairs  of  corresponding  values  of  two  variable  quantities. 
Frequently  these  values  are  found  by  experiment,  and  the 
general  law  which  they  satisfy  may  be  known  or  suspected. 
The  following  illustrations  will  show  how,  in  some  of  the 
simpler  cases,  the  law  may  be  tested  and  the  constants  of  the 
equation  determined. 

The  more  difficult  problems  of  this  nature  can  be  treated  by 
the  use  of  Fourier's  series,  a  method  of  wide  application,  but 
too  diffi,cult  to  discuss  here. 

197.  Points  lying  on  a  straight  line.  The  simplest  case  that 
occurs  is  that  where  the  points  whose  coordinates  are  the  two 
measured  quantities  lie  on,  or  approximately  on,  a  straight 
line.  In  this  case  one  has  only  to  select  the  straight  line 
which  seems  to  best  fit  the  points,  and  write  its  equation. 
The  equation  of  this  line  is  then  the  equation  connecting  the 
variables  if  the  same  scale  has  been  used  throughout.  In 
plotting  the  points,  however,  any  convenient  scales  may  be 
used,  and  the  equation  of  the  line  written  with  any  other  scale 
that  is  desired.  The  two  coordinates  in  the  equation  of  the 
line  must  then  be  expressed  in  terms  of  the  two  variables  be- 
tween which  an  equation  is  sought.  The  substitution  of  these 
values  in  the  equation  of  the  line  gives  the  desired  equation. 

Example.     The  extension  of  a  certain  wire  when  loaded 

223 


224 


ANALYTIC  GEOMETRY 


was  observed  to  be  as  shown  in  the  following  table,  where  E  is 
the  elongation  in  inches,  and  W  is  the  load  in  pounds. 

TT       1         2        3        4        5        7        10        12        15        18 
E     .12      .23      .34      .46     .58      .80     1.16      1.39     1.74     2.09 


^  ^  ..-.".-  .1    .11   .11 .....      ."Z'-  -   ... 

"                                                                  <<\\ 

O           .       .           .  .       .                .       .             .   .;»' 

f                                                  L>rT 

"     :::::::::::     ::  i?^: :::     :::     ::  ::: 

■^      '.".'.".'. '.2'  "  " " ' 

j^Pn 

'it*           '        '      "     -    -  -   -      ■ 

-^f  S :..::::.:..:::::::::::::: 

5  10  15 

w.  2  spaces  to  1  lb. 

Fig.  147. 


20 


On  plotting  the  points  whose  coordinates  are  the  correspond- 
ing values  of  E  and  W,  they  are  seen  to  lie  approximately  on  a 
straight  line.  The  line  which  seems  to  best  fit  the  points 
passes  through  the  origin  and  the  point  (38,  22).     The  equa- 

22 

tion  of  this  line  is  therefore  i/  =  —  a?.     But  in  the  scale  used, 

^38 

x  =  2Wyy  =  10E,  and  hence  the  equation  connecting  E  and 


TFislO^  =  — TT,  or 


E  =  .116  W. 


This  equation  therefore  holds  approximately' for  the  particular 
wire  used  and  within  the  limits  of  the  observed  values. 

Exercise.     From  the  following  corresponding  values  of  u 
and  V  determine  the  equation  connecting  them. 

u       1    1.5    2.3    3.1      3.8      4.2      5.0      5.8      6.5      7.2      8.0 
V    5.5    6.4    8.2    9.7    11.0    11.9    13.5    15.0    16.5    18.0    19.5 


EMPIRICAL  EQUATIONS 


225 


198.   The  curve  y  =  Cac\      A  number   of  curves   obtained 
from  physical  measurement  follow  the  law  y  =  Cic**. 

If  the  logarithm  of  both  sides  of  the  equation  be  taken,  there 
results 

log  y  =  log  C-\-n  log  x. 

If  now  u  =  log  Xj  v  =  log  y,  b  =  log  C,  then 
v  =  b-\-  nu. 
This  is  an  equation  of  first  degree  in  u  and  v.     Hence  if  u  and 
V  be  taken  as  coordinates  and  the  points  representing  corre- 
sponding values  plotted,  these  points  will  lie  on  a  straight  line. 

Conversely,  if  the  points  (log  x,  log  y)  do  lie  on  a  straight 
line,  the  equation  of  the  line  is  of  the  form 

v=:nu-\-b,  where  u  =  log  x,  v  =  log  y ; 
i.e.  log  2/  =  w  log  X  +  log  (7,  if  6  =  log  C, 

or  log  y  =  log  (Ca?**). 

y=  Cx^. 


2.S00 


2L000 


1.600 


1.400 


L200 


^^ 


^^^ 


1.00  L20 


Fia.  148. 


226  ANALYTIC  GEOMETRY 

The  following  illustration  will  show  how  the  constants  C 
and  n  may  be  determined  when  the  points  lie  on  a  curve  of 
this  kind. 

Illustration.  The  following  represent  pressure  p  and  vol-, 
ume  V  oi  Si  gas : 

V       3  4  5.2        6.0        7.3        8.5      10 

p  107.3      71.5      49.5      40.5      30.8      24.9      19.8 

Let  X  =  log  V,  y  =  log  p.     Then  the  values  of  x  and  y  are 
X     ATT         .602        .716         .778        .863         .929       1.000 
y  2.031       1.854      1.695      1.607       1.489       1.396      1.297 

The  points  determined  by  x  and  y  are  seen  to  lie  on  a  straight 
line,  approximately,  Fig.  148.  The  slope  of  this  line  is  found 
by  measurement  to  be  —  |f,  or  — 1.40.  Then,  since  the  line 
passes  through  (1,  1.297),  its  equation  is 

y  -  1.297  =  - 1.40  (x-1), 
or  2/ =  -1.40  a; +  2.697. 

But  2.697  =  log  497.7. 

Therefore,  since  y  =  logp,  x  =  log  v, 

logp  + 1.40  log  V  =  log  497.7, 
or  i)?;i''«  =  497.7, 

which  is  therefore  approximately  the  formula  connecting  p 
and  V. 

The  correctness  of  this  formula  should  be  tested  by  substi- 
tuting some  or  all  of  the  values  otp  and  v  in  the  given  table. 

E.g.  if  V  =  5.2  and  p  =  49.5, 

then  log 'y=. 7160, 

which  multiplied  by  1.40  gives 

log  5.2i'«>=  1.0024 

log  49.5  =  1.6946 

logpvi«=:  2.6970 

pvi*^  =  497.7, 

which  checks  the  result  already  found. 


EMPIRICAL  EQUATIONS  227 

Where  the  points  do  not  lie  so  accurately  on  the  line  as  in 
this  example,  it  would  be  better  after  obtaining  the  slope  of 
the  line  to  write  pv"  =  C,  and  having  n,  substitute  the  given 
values  of  p  and  v  to  find  C.  Make  this  computation  for  each 
pair  of  values  given,  and  take  the  average  of  the  values  found 
for  C. 

Exercise.  Find  the  equation  connecting  Q  and  Ji  from  the 
following  observed  values. 

h     .583        .667        .750        .834        .876        .958 
Q  7.00        7.60        7.94        8.42        8.68        9.04 

199.  The  curve  y  =  ab^,  or  y  =  ae^%  where  e  =  2.71828  .... 
Certain  physical  quantities  are  connected  by  an  equation  of 
the  form  y  =  ah'  where  a  and  h  are  constant.  If  it  is  thought 
that  two  quantities  for  which  several  corresponding  values  are 
known  obey  this  law,  they  may  be  tested,  and,  if  the  law  is 
fulfilled,  the  values  of  the  constants  determined  as  follows: 
Plot  the  points  whose  abscissas  are  x  and  whose  ordinates  are 
log  2/.  If  they  lie  on  a  straight  line,  the  supposed  equation  is 
correct,  otherwise  not.     This  follows  from  the  fact  that  if 

y=ah%  (1) 

then  log  2/ =  log  a  +  a;  log  6,  '  (2) 

and  the  converse. 

Suppose  the  points  («,  logiy)  lie  on  a  straight  line.  The 
slope  of  this  line  is  then  the  value  of  log  h  (see  eq.  (2)),  from 
which  h  may  be  found.  Also  the  intercept  of  the  line  on  the 
axis  of  ordinates  is  log  a.  From  this  intercept  a  may  then  be 
found.     However,  it  will  be  more  accurate  to  obtain  a  from 

the  average  of  the  values  of  ^  after  h  is  determined  from  the 
line. 

In  some  cases  a  is  1,  and  this  will  be  indicated  by  the 
straight  line  passing  through  the  origin. 

If  it  is  desired  to  express  y  =  ah''  in  the  form  y  =  ae^,  one 


228 


ANALYTIC  GEOMETRY 


has  only  to  let  h  =  e^,  for  then  6*=(e*)'=  =  e*''.     To  determine  h^ 

log6_  log& 
log  e~. 4343* 


log  h  =  k  log  e,  or  k 


Example.     The  values  of  x  and  y  of  the  following  table  are 
thought  to  be  connected  by  an  equation  of  the  form  y  =  ah'. 
X   2  3.2  4.7  8.5 

y  7.086        12.64        26.07        163.0 

Form  then  the  following  table : 
X  2  3.2  4.7  8.5 

logy     .8504      1.1017      1.4161      2.2122 

Plot  the  points  (a?,  log  y).     They  are  seen  to  lie  on  a  straight 
line. 


10.3 

12.6 

388.4 

1178 

10.3 

12.6 

2.5893 

3.0711 

A 

/. 

^ 

y 

Q^ 

^ 

-^                     ^- 

/ 

^             ^"^    - 

y 

1             J^                 - 

J^ 

T 

/ 

0                        5          , 

10                   15 

Fig.  149. 


The  slope  of  this  line,  computed  by  using  the  extreme  values 
of  X  and  log  y  in  the  table,  is 


Hence 


3.0711  -  .8504  ^  2.2207 
12.6-2  10.6 

log  h  =  .2095, 
b  =  1.62. 


.2095. 


EMPIRICAL  EQUATIONS  229 

To  determine  a,  a  =  ^  • 

log a  =  logy  —  x  log  b 
=  log  2/  -  .2095  X. 

Using  X  and  logy  from  the  table  of  values,  the  following 
values  of  log  a  are  obtained. 

.4314,     .4313,     .4314,     .4314,     .4314,     .4314. 

The  average  of  these  is 

log  a  =  .4314. 
.-.  a  =  2.70. 
.-.2/ =2.70(1.62'). 

200.  Some  special  substitutions.  In  some  other  cases,  if  the 
law  connecting  the  variables  is  suspected,  the  correctness  of 
the  supposition  may  be  easily  tested  by  a  substitution  which 
will  reduce  the  problem  to  that  of  the  straight  line. 

For  example,  if  it  is  thought  that  the  relation  is  y  =  a-\-—, 

of 

plot  the  points  ( -^j  2/ )  •     If  these  points  lie  on  a  straight  line, 

the  assumed  equation  is  correct,  and  the  quantities  a  and  b  can 
be  found  from  the  graph. 

In  like  manner  the  equation  xy  =  ax-\-  by,  an  hyperbola, 
may  be  written 

y  =  a-\-by.,  (1) 

X 


or 


or 


x  =  b  +  a'^,  (2) 

y 

1  =  -  +  -,  (3) 

X      y 

and  these  may  be  reduced  to  the  straight  line  form  by  using 

u  for  ^  in  (1),  u  for  -  in  (2),  and  u  and  v  for  =  and  -  in  (3). 


230  ANALYTIC  GEOMETRY 

ExERcise.  Prove  that  the  following  points  lie  on  a  curve 
of  the  form  xy  =  ax  +  by,  and  determine  a  and  b. 

X           1.59          1.96        2.27        3.12        5.00          7.15  16.7 

y             .885        1.11        1.28        1.85        3.24         5.10  22.0 

201.  The  curve  y  =  a-^bac  +  cx^  +  da^  +  •  +  kx\  When 
no  other  equation  can  be  found  to  fit  the  given  points  the 
equation 

y  =:  a -\- bx -^  cx^ -\-  dx^  +•••-!-  Jcx"" 

may  be  assumed,  and  by  substituting  the  coordinates  of  the 
given  points  enough  equations  can  be  obtained  for  the  deter- 
mination of  the  constants  a,  b,  c,  •  •  •  1c. 

The  number  of  terms  to  assume  will  depend  upon  the  num- 
ber and  location  of  the  given  points.  If  the  curve  on  which 
the  points  lie  diverges  only  slightly  and  in  one  direction  from 
a  straight  line,  it  will  usually  be  sufficient  to  assume  three 
terms  on  the  right.  This,  of  course,  makes  the  curve  a  pa- 
rabola. But  each  case  must  be  settled  on  its  merits,  and  the 
construction  of  the  curve  from  the  equation  which  Is  found 
will  be  the  test  of  the  accuracy  with  which  it  fits  the  given 
points. 

Example.  To  find  the  equation  of  a  curve  through  the  fol- 
lowing points: 

X  8        23        39        53        63 

y        10        19        27        33        36 

These  points  when  plotted  are  seen  to  lie  on  a  curve  which 
resembles  a  portion  of  a  parabola  with   axis  parallel  to  the 
2/-axis.     It  is  worth  while  then  to  try 
y  =  a-{-bx-\-  coi?. 

Take  the  two  extreme  points  and  the  middle  point  for  the  de- 
termination of  the  coefficients.     The  equations  obtained  are 

10  =  a-|-  86+  64c, 
27  =  a-}-39&-f-1521c, 
36  =  a  +  63  6  +  3969  c. 


EMPIRICAL  EQUATIONS  231 

Solving  these  equations, 

a  =  4.63,         6  =  .697,        c= -.00315. 
Hence  the  approximate  equation  is 

y  =  4.63  +  .697  x  -  .00315  a^. 

The  substitution  of  the  intermediate  values  of  x  not  used  in 
the  computation  of  a,  6,  and  c  give, 

for  a;  =  23,  y  =  l^M, 

forx  =  53,  2/ =  32.72, 

which  are  reasonably  close  to  the  values  of  19  and  33. 

If  greater  accuracy  is  desired,  four  or  five  terms  may  be 
assumed  on  the  right  and  then  four  or  five  of  the  given  points 
used  to  determine  the  constants. 

Again,  different  seta  of  the  given  points  might  be  used  to 
determine  the  constants  and  average  values  of  the  constants  so 
found  used. 

EXERCISE  XLH 

1.  In  an  experiment  to  determine  the  deflection  of  a  beam  of  varying 
length  the  following  measurements  were  made  : 

Length  (in.)      12  16         20  24  28         32  36  40 

Deflection  (in.)  .017       .043       .085       .145       .220      .342       .512         .713 

Prove  that  the  deflection  d  and  the  length  L  are  connected  by  an 
equation  of  the  form 

d  =  CL», 
and  find  the  values  of  m  and  C. 

2.  Find  an  equation  connecting  x  and  y  to  fit  the  following  values  : 
X  .6  1.2         1.6         2.2         2.8         3.4  4.3  6.0 
y          .801       1.70       2.54       3.98       5.58       7.32       10.17       16.22 

3.  Prove  that  the  following  values  of  x  and  y  satisfy  an  equation  of 
the  form 

1  -\-  bx 
and  find  the  values  of  a  and  6. 

x  .5  1.2  2.0  3.4  4.1  5.3 

y        1.08        2.41        3.71        5.56        6.33        7.46 


232  ANALYTIC  GEOMETRY 

4.  The  following  numbers  are  taken  from  a  table  : 

X        1.1         1.4        2.0        2.6        3.4        4.1        6.3        7.8        9.8 
y  .095      .336       .693       .956     1.224     1.386     1.841     2.054    2.282 

Find  the  equation  connecting  x  and  y. 

Suggestion.     Plot  the  points  (log  x,  y'). 

5.  Prove  that  the  following  values  of  u  and  v  satisfy  an  equation  of 
,the  form  v  =  a -\ ,  and  find  the  values  of  a  and  6: 

M 

V 

6.  Find  an  equation  to  fit  the  following  values  ot  p  and  y; 

(Trypv"  =  C.) 

V        4.2  4.7  5  5.5  6.2  7  8  9 

p        105        92  86        78  68  60        53        46 


.5 

1.1 

1.7 

2.3 

5.1 

6.4 

13.6 

4.00 

2.37 

1.84 

1.33 

1.28 

ANALYTIC   GEOMETRY   OF   SPACE 


CHAPTER  XVII 


COORDINATES  IN  SPACE 


202.  Rectangular  coordinates  in  space.  As  on  a  straight 
line  one  quantity  was  required  to  determine  the  position  of  a 
point,  and  in  the  plane  two  quantities,  so  in  space  three  quanti- 
ties are  necessary.  One  way  of  choosing  these  quantities  is 
the  following :  Through  any  point  0,  chosen  as  an  origin,  draw 
three  mutually  perpendicular  lines  OX,  0  Y,  OZ.  These  lines 
determine  three  mutually  perpendicular  planes  XY,  XZ,  YZ. 
From  any  point  P  in  space  let  perpen- 
diculars be  drawn  to  the  three  planes. 
Then  the  distances  measured  from  the 
planes  to  the  point  are  called  the  rec- 
tangular coordinates  of  the  point  P. 

Let  distances  measured  in  the  direc- 
tion of  OX,  OY,  and  OZ,  i.e.  to  the 
right,  forward,  and  upward,  be  counted 
as  positive,  and  distances  in  the  oppo- 
site direction,  i.e.  to  the  left,  backward, 
and  downward,  as  negative.  Then  to 
every  set  of  three  real  numbers  there  corresponds  a  point  in 
space  and  conversely. 

The  distances  SP,  QP,  and  NP  (Fig.  150)  are  called  respec- 
tively the  Qc,  2/,  and  s  of  the  point  P,  and  the  point  is  denoted 
t>y  (x,  y,  z),  or  by  P(x,  y,  z). 

The  plane  containing  OX  and  01^  is  called  the  a?2/-plane,  and 
similarly  for  the  others. 

The  three  planes  containing  the  axes  are  known  as  coordi- 
nate planes. 

233 


z 

Q 

/ 

/ 

p 

M 

X 

/ 

0 

/ 

N 
Fig.  150. 


234  ANALYTIC  GEOMETRY 

The  eight  portions  of  space  separated  by  the  coordinate 
planes  are  called  octants. 

Two  points  are  said  to  be  symmetric  with  respect  to  a  plane 
when  the  line  joining  the  points  is  perpendicular  to  the  plane 
and  is  bisected  by  it. 

EXERCISE  XLin 

1.  Locate  the  points  (1,  3,  2),  (-1,  3,  4),  (1,  -  2,  4),  (1,  3,  -  2), 
(2,  -  3,  -  4),  (-  1,  -  2,  -  3),  (-  1,  -  2,  3),  (-  1,  3,  -  2),  (0,  1,  2), 
(2,0,0),  (0,0,0). 

2.  Show  that  the  line  OP  in  Fig.  150  is  the  diagonal  of  a  rectangular 
parallelopiped  of  which  the  numerical  values  of  x,  y,  and  z  are  the  lengths 
of  the  sides. 


3.  Show  that  OP  =  Vx^  +  y'^  +  z'^- 

4.  Find  the  distance  from  the  origin  to  each  of  the  points,  (1,  3,  —  2), 
(3,-1,4),  (2,-1,  -3). 

5.  Find  the  point  symmetric  to  each  of  the  following  points  with 
respect  to  each  of  the  coordinate  planes,  (2,  3,  4),  (—3,  —  1,  —  2), 
(3,  -  1,  2). 

6.  Find  the  point  symmetric  to  each  of  the  following  points  with 
respect  to  the  origin,  (2,  3,  5),  (-  2,  4,  3),  (3,  -  4,  -  1). 

7.  Prove  that  (a,  &,  c)  and  (—a,  —  6,  —  c)  are  symmetric  with 
respect  to  the  origin. 

8.  What  is  the  value  of  x  for  any  point  in  the  i/sj-plane  ?  What 
therefore  is  the  equation  of  the  y^-plane  ?  What  are  the  equations  of  the 
other  coordinate  planes  ? 

9.  Where  do  all  points  lie  that  have  a;  =  0  and  y  =  0?  What  are 
the  equations  of  the  coordinate  axes  ? 

10.   Find   the   locus  of   points  which   satisfy  the  following  sets   of 
conditions : 

(a)  x  =  y,z  =  0.  (/)  x  =  2,y  =  3. 

(b)  x  =  y,z  =  2.  (g)  x^  +  y^  =  16,  ^  =  0. 

(d)  x  =  y  =  z.  ^^^  a^-^b^-   '''-    ' 

(e)  —x  =  y,y  =  z.  (i)  y^  =  4x,z  =  S. 


COORDINATES  IN  SPACE 


235 


203.  Distance  between  two  points  in  rectangular  coordi- 
nates. Let  the  points  be  Pi(xi,  y^,  z^  and  P2(^2j  2^2?  ^2)- 
Through  P^  and  P^  pass  planes  parallel  respectively  to  the 
three  coordinate  planes.  These  three  planes  form  a  rectan- 
gular parallelopiped  of  which  P1P2  is  the  diagonal,  and  the 
edges  are  respectively  the  differences  of  the  coordinates  parallel 
to  the  edges. 

Thus,  in  Fig.  151,  PiiV=  x^  —  x-^,  NM=  y^  —  2/1,  MP^  =  z^—  z^. 


But 


P^P\  =  PiAt'  4-  nm"  -{-MPI 


',  d  =  r,r,  =  V(p,,  -  ^,Y + (2/1  -  2/2)' + («i  -  «2)'. 


Fia.  151. 


If  the  two  points  are  the  origin  and  the  point  (a;,  y^  2),  this 
formula  becomes 


204.  Point  dividing  a  line  in  a  giveii  ratio.  If  the  point 
(a;,  y,  z)  divides  the  line  from  (a^j,  2/1,  z^  to  (a^g,  2/2?  ^2)  i^i  the  ratio 
r :  1,  then 

^^^H-r^^  y^ih±nh  ^^^i  +  yg2 

1  H-r    '  1  -t-r*   '  \-\-r 

The  proof  is  left  to  the  student. 


fxi  + 


236  ANALYTIC  GEOMETRY 

EXERCISE  XLIV 

1.  Find  the  distance  between  (3,  4,  —  2)  and  (—  6,  1,  —  6). 

2.  Prove  that  the  center  of  gravity,  i.e.  the  intersection  of  the  medians 
of  tlie  triangle  whose  vertices  are  (xi,  yi,  Zi),  {X2,  2/2,  ^2)*  and  (0:3,  y^,  z^), 

x^  +  xs  ■  yi  +  yo-h  y3    £i_±_52_±£i\ 
3  '  3  '  3  j' 

3.  Show  that  the  lines  drawn  from  the  vertices  of  a  tetrahedron  to  the 
intersection  of  the  medians  of  the  opposite  faces  meet  in  a  common  point 
which  is  I  the  distance  from  each  vertex  to  the  opposite  face. 

4.  Write  the  equation  which  expresses  the  condition  that  (x,  y,  z) 
shall  be  equidistant  from  (0,  0,  0)  and  (3,  5,  1).  What  is  the  locus  of 
(x,  y,  z)  ? 

5.  Write  the  condition  that  (x,  y,  z)  shall  remain  at  the  distance 
4  from  (0,  0,  0) .     What  is  the  locus  of  (x,  y,  z)  ? 

6.  Find  the  equation  of  the  surface  of  a  sphere  with  center  at 
(2,  1,  —  3)  and  radius  5. 

205.  Polar  coordinates.  A  point  in  space  may  be  deter- 
mined, by  its  distance  from  the  origin  and  the  angles  which  the 

line  from  the  origin  to  the  point 
makes  with  the  rectangular  coordi- 
nate axes. 

Thus,  let  OX,  OY,  OZ,he  a  set  of 
rectangular  axes,  and  let  P  be  any 
point  in  space.  Then  OP  and  the 
angles  a,  jS,  y,  between  OP  and  the 
axes  of  X,  y,  and  z,  respectively,  de- 
termine   the    position    of    P.      If 

Fig.  152.  ^  ^  , ,  .    ,  1        j         -    j 

OP=:r,  the  point  may  be  denoted 
by  (r,  a,  p,  y).  The  four  quantities  r,  a,  fB,  y  are  sometimes 
called  the  polar  coordinates  of  P. 

It  is  convenient  to  restrict  r,  a,  ^,  y  to  positive  values,  and 
to  further  restrict  the  angles  to  values  not  greater  than  180°. 
Any  point  in  space  may  be  represented  by  such  values  of 
r,  a,  ^,  y. 


COORDINATES  IN  SPACE 


237 


The  angles  a,  jS,  y  are  called  the  direction  angles  of  OP,  and 
the  cosines  of  these  angles  the  direction  cosines  of  OP. 

206.   Relation  between  rectangular  and  polar  coordinates  of 
a  point.     From  Fig.  153,  if  the  rectangular  coordinates  of  P 
are  x,  y,  z,  then  the  following  rela- 
tions are  seen  to  hold: 
0?  =  r  cos  a, 
2/  =  r  cos  p, 
z  =  rco»y. 


Since  r  =  Vic^  +  2/^  +  ^^  the  above 
equations  may  be  solved  for  the 
direction  cosines  and  the  following 
values  obtained: 


a  = 


cc 

V^ 

+  z^ 

v^ 

+  2/'^ 
z 

+  z^ 

Fig.  153. 


cosp  = 
COS  7  = 

207.  Relation  between  the  direction  cosines  of  a  line. 

Definition.  The  direction  cosines  of  a  given  directed  line 
are  the  direction  cosines  of  a  line  drawn  from  the  origin  in  the 
same  direction  as  the  given  line. 

If  the  three  equations  of  the  preceding  article, 
a?  =  r  cos  a, 
y  =rG0s/3f 
z  —  r  cos  y, 

be  squared  and  added,  there  is  obtained 

x^ -\- y"^  -\- z^  =  T^  (cos^  a  +  cos^  ^  +  cos^  y). 
But  ar'  +  y^  +  ^^^r^. 

.-.  cos2  a  +  cos2  p  +  cos2  7  =  1. 


238 


ANALYTIC  GEOMETRY 


Hence,  the  sum  of  the  squares  of  the  direction  cosines  of  any 
straight  line  is  1. 

208.   Direction  cosines  of  a  line  joining^  two  points.      Let 

^i(^ij  2/ij  ^i)  aiid  P2(^2>  2/2?  ^2)  be  any  two  points  in  space  and 
consider  the  line  as  directed  from  P  to  Pg- 


Fig.  154. 


Let  the  direction  cosines  of  PjPg  ^6  cos  a,  cos  p,  cos  y. 


Then 


cos  a 


_  a?2  —oci 


^.-P=^^--v 


^2  —  gl 


where  d  =  V  (a^  -  aja)^  +  (2/1  -  2/2)'  +  (%  -  2!2)^- 

These  relations  are  evident  from  Fig.  154. 

209.    Spherical  coordinates.     Again,  take  the  three  mutually- 
perpendicular  axes  OX,  OY,  OZ. 

Let  P  be  any  point  in  space.     Then  the  position  of  P  is 

determined  by  the  distance  r,  or 
OP,  and  the  angles  ^  and  <^,  where 
6  is  the  angle  between  OP  and  the 
positive  OZ,  and  <^  is  the  angle  be- 
tween the  positive  OX  and  the 
orthogonal  projection  of  OP  upon 
the  a72/-plane. 

The  point  is  denoted  by  (r,  0,  <f>). 

The  quantities  r,  6,  and  <^  are 

called  the  spherical  coordinates  of 

P. 

Fig.  165.  The  student   can   easily   show 


COORDINATES  IN  SPACE  239 

that  if  P  has  rectangular  coordinates  {x,  y,  z),  then  the  relations 
between  the  rectangular  and  spherical  coordinates  of  the  point 
are 

ac  =  r  sin  6  cos  <|), 

y  =  rsmQ  sin  <j), 
z  =  rcosQ. 

Spherical  coordinates  are  useful  in  some  surveying  and  as- 
tronomical problems. 

EXERCISE  XLV 

1.  Find  the  direction  cosines  of  the  line  from  the  origin  to  (2,  —1,3). 

2.  Show  that  if  any  three  real  quantities,  a,  6,  c,  be  chosen,  a  line 
With  direction  cosines  proportional  to  these  quantities  can  be  found,  and 

that  the  direction  cosines  are  -,  -,  - ,  where  d  =  Va^  +  &2  _^  ^2^ 
d    d    d 

3.  Find  the  direction  cosines  of  the  line  from  (3, 1,  —  2)  to  (—  1,  4,  3). 
Draw  the  figure. 

4.  Given  cos  a  =  ^,  cos  jS  =  ^,  find  cos  7. 

5.  Find  the  rectangular  coordinates  of  a  point  whose  polar  coordinates 
are  (2,  30°,  45°,  7).     How  many  solutions  ? 

6.  Find  the  spherical  coordinates  of  a  point  whose  rectangular  coordi- 
nates are  (3,  2,  4). 

7.  Find  the  spherical  coordinates  of  a  point  in  terms  of  the  rectangular 
coordinates  of  the  point. 

8.  Show  that  reversing  the  direction  of  a  line  changes  the  sign  of  each 
direction  cosine. 

9.  Write  the  direction  cosines  of  each  coordinate  axis. 

210.  Projection  of  a  line  upon  another  line. 

Definition.  From  the  extremities  A  and  5  of  a  line  AB 
drop  perpendiculars  upon  a  line  MN,  meeting  it  in  C  and  D 
respectively.  Then  CD  is  called  the  orthogonal  projection  of 
AB  upon  MN.     (Fig.  156.) 

Only  orthogonal  projection  will  be  used  in  what  follows, 


240 


ANALYTIC  GEOMETRY 


and  projection  will  be  understood  to  mean  orthogonal  pro- 
jection. 

Definition.  The  angle  between  two  non-intersecting  lines 
is  defined  to  be  the  angle  between  two  intersecting  lines  drawn 
in  the  same  directions  respectively  as  the  given  lines. 


D     N  N    C 

Fig.  156. 

If  a  is  the  angle  between  AB  and  JOT",  and  I  is  the  length  of 
ABy  then 

?  cos  a  =  projection  of  AB  on  M'S'. 

Proof.  Through  B  pass  a  plane  perpendicular  to  MN  and 
through  A  draw  a  line  parallel  to  MN  to  cut  this  plane  in  E. 
(Fig.  156.) 

Then  AE=CD. 

But  AE  =  I  cos  a. 

.'.  CD  =  I  cos  a. 


(If  a>90°,  CD  is  negative,  i.e.  is  opposite  in  direction  to 

211.  Projectioii  of  a  broken  line.  The  projection  on  any 
axis  of  a  straight  line  joining  two  points  is  equal  to  the  sum  of 
the  projections  on  the  same  axis  of  the  sides  of  any  broken 
line  connecting  the  two  points,  if  the  parts  of  the  broken  line 
are  directed  so  that  the  beginning  of  each  side  after  the  first  is 
at  the  end  of  the  preceding. 

This  is  evident  from  the  definition  of  projection. 


COORDINATES  IN  SPACE 
Thus  in  Fig.  157, 


241 


Fig.  157. 

ab  =  ac -\- cd -^  de -\- ef  +  fb, 
or  proj.  AB  =  proj.  AG -\- -pro j.  CD  +  proj.  DE  +  proj.  EF 

+  proj.  FB. 

If  I,  li,  I2,  "•  ?5  are  the  lengths  of  AB,  AC,  CD,  •••  FB,  respec- 
tively, and  a,  a^,  a^,  •••  a^  are  the  angles  between  these  lines 
and  MN,  then 

I  cos  a  =  Zi  cos  «!  +  ?2  cos  a^-\-  •  ••  +  Z5  cos  ag. 

212.  The  angle  between  two  lines  in  terms  of  their  direction 
cosines.  Let  two  lines  have  direction  angles  «i,  ySi,  yi,  and  rtg? 
Aj  72)  respectively,  and  let  0  be  the  angle  between  them.  To 
find  the  value  of  Q. 

Through  the  origin  draw  two  lines  OPj  and  OP2  having  the 
same  directions  respectively  as  the  two  given  lines. 

Let  the  coordinates  of  Pi  be  {x^,  y^,  z-^  and  let  OP^  =  i\. 

On  OP.  project  OP^  and  the  broken  line  OM-^MN-\-NP^ 
(Fig.  158).     Since 

proj.  OPi=proj.  Oitf  +  proj.  JOT  +  proj,  iVPj, 

therefore, 

7*1  cos  d  =  X]^  cos  a2  +  2/1  cos  /Sa  +  «i  cos  y^ 


242 

But 

or 


ANALYTIC  GEOMETRY 

Xi  =  riCosa.,  2/1  =  ^1  COS  ySu  2;i  =  riCosyi. 
fi cos  0  =  riCCa  Ui cos  02  +  ^1  cos  I3i cos p2  4-  ^1  cos  yi  cos  yaj 
cos  6  =  cos  tti  cos  a2  +  cos  pi  cos  P2  +  cos  ^i  cos  'yg* 


_/^ 

y 

f 

^ 

X 

..H 

x 

Yi 

M 

t 

w 

k 

Fig.  158. 

(Notice  that  if  one,  or  more,  of  the  coordinates  x^,  y^,  z^  is 
negative,  e.g.  y^,  then  —  ?/i  is  the  length  of  MN,  but  180°  —  ySj 
is  the  angle  between  MN  and  OP2 ;  hence  the  middle  term  is 
—  2/1  cos  (180°  ~  ^2)?  which  is  the  same  as  2/1  cos  ft-) 

EXERCISE  XL VI 

1.  Find  the  projection  of  the  line  from  (2,  1,  —  3)  to  (3,  —  4,  5)  upon 
each  of  the  coordinate  axes. 

2.  The  direction  cosines  of  a  line  are  proportional  to  2,  3,  and  —  4. 
Find  their  values. 

3.  Express  in  terms  of  the  direction  cosines  of  two  lines  the  condition 
that  the  two  lines  be  parallel.    The  condition  that  they  be  perpendicular. 

4.  Find  the  angle  between  two  lines  whose  direction  cosines  are 
respectively  proportional  to  2,  —  1,  3  and  1,  3,  —2. 


CHAPTER  XVIII 

LOCI   AND   THEIR   EQUATIONS 

213.  Certain  straight  lines  and  planes.  The  student  has 
already  considered  some  simple  equations  of  straight  lines  and 
planes.  For  example,  a;  =  a  is  the  equation  of  a  plane  parallel 
to  the  2/2-plane. 

The  two  equations  y  —  h,  z  =  c,  represent  a  straight  line  par- 
allel to  the  ic-axis,  the  intersection  of  the  two  planes  y  =  b  and 
z  =  c. 

The  two  equations  x  =  y,  z  =  c,  represent  a  straight  line,  the 
intersection  of  the  plane  z  =  c  and  a  plane  bisecting  the  dihe- 
dral angle  between  the  iC2;-plane  and  the  yz--p\sine. 


214.   Cylinders  with  elements  parallel  to  a  coordinate  axis. 

Consider  a  circular  cylinder  with  the  2-axis  for  its  axis  and 
with  radius  r.     (Fig.  159.) 

If  any  point  P  be  taken  on  the  sur- 
face of  this  cylinder,  the  x  and  y  of 
the  point  are  the  same  as  the  x  and 
y  of  the  projection  of  the  point  on  the 
ic?/-plane.  But  these  latter  values 
satisfy  the  equation  of  the  circle 
x^-\-y^  =  7^.  Hence  the  coordinates 
of  P  satisfy  the  same  equation. 

The  equation  of  the  surface  of  the 
cylinder  is  therefore 

of  -\-y^z=  T^, 

In  like  manner  it  may  be    shown 
that  if  a  straight  line,  kept  always  parallel  to  the  2;-axis,  is 
moved  along  any  curve  in  the  aji^-plane,  a  cylindrical  surface  is 

243 


z 

X 

\ 

^ 

? 

X 

X 

k 

w 

Fig.  159. 


244 


ANALYTIC  GEOMETRY 


generated  which  has  the  same  equation  as  the  equation  of  the 
curve  in  the  xy-^\sine. 

Thus  the  equation  y^  =  4:X,  interpreted  as  an  equation  of  a 
locus  in  space,  is  the  equation  of  a  cylindrical  surface  generated 
by  a  straight  line  parallel  to  the  2;-axis,  moving  along  the  curve 
y^z=4:xin  the  a^^z-plane. 

Likewise  an  equation  of  the  form  y  =  f(z),  read  "1/  equals/ 
of  0,"  i.e.  ?/  is  a  function  of  z,  is  the  equation  of  a  cylindrical 
surface  generated  by  moving  a  line  parallel  to  the  ic-axis  along 
the  curve  y=zf{z)  in  the  2/2-plane. 

The  student  should  describe  the  locus  in  space  of  the  equa- 
tion z=f(x). 

EXERCISE  XL VII 

Describe  and  sketch  the  loci  in  space  of  the  following  equations ; 

1.  ic2  +  02-25.  5.    x^  =  2pz. 

2.  (x  -  ay  +(2/  -  &)2  =  r2.  Q    ^^t=l, 

3.  ic  cos  a  +  ?/  sin  a  =p,  «"      ^'^ 

7     w2  _  ^2  _  0,2 


^  =  1. 


8.   ?/  =  mz  -f-  c. 


215.   Surfaces  of  revolution.     If  the  equation  of  a  curve  in 

one  of  the  coordinate  planes  is 
known,  the  equation  of  the  sur- 
face formed  by  revolving  this 
curve  about  one  of  the  coordinate 
axes  can  be  obtained  from  it. 

As  an  illustration,  consider 
the  surface  formed  by  revolving 
about  the  aj-axis  the  parabola 
2/2  =  4  a;. 

Let  P{x,  ?/,  z)  be  any  point  on 
this  surface.     Then  (Fig.  160) 
x=OM,y=  MN,z  =  NP. 
Since  MP=  MR,  it  follows  from  the  equation  of  the  parabola 


LOCI  AND  THEIR  EQUATIONS  245 

that 

But  Mp  =  MN^'-^Np^f^-^. 

This  is  therefore  the  equation  which  is  true  for  any  point  on 
the  surface,  and  clearly  for  no  other  points,  and  hence  is  the 
equation  of  the  surface. 

The  process  of  obtaining  the  equation  of  the  surface  from 
that  of  the  curve  in  the  xy-^\di,nQ  consists  in  replacing  y  by 

In  general,  if  any  curve  in  the  a;i/-plane,  F(x,  y)  =  0,  be  re- 
volved about  the  aj-axis,  the  equation  of  the  surface  formed  is 


F(x,Vy'-bz')=0. 

EXERCISE  XLVin 

1.  Find  the  equation  of  the  surface  generated  by  revolving  the  curve 
y^  =  4:X  about  the  y-axis.     Sketch  the  figure  in  one  octant. 

2.  Find  the  equation  of  the  surface  generated  by  revolving  the  circle 
x^  -j-y^  =  r^  about  the  ic-axis  ;  about  the  y-&xis. 

3.  Find  the  equation  of  the  surface  of  the  spheroid  generated  by  re- 
volving  the  ellipse  — \-^  =  1  about  the  y-axis.     The  spheroid  is  said  to  be 

oblate  if  a  >  6,  prolate  if  a  <  6. 

4.  Find  the  equation  of  the  surface  of  a  cone  generated  by  revolving 
the  line  y  =  mx  about  the  x-axis. 

216.   Nature  of  locus  determined  by  plane  sections.     It  is 

frequently  useful,  in  trying  to  determine  the  nature  of  a  locus, 
to  find  the  intersection  of  the  locus  by  a  plane.  Generally  the 
planes  parallel  to  the  coordinate  axes,  or  else  containing  a 
coordinate  axis,  are  the  simplest  ones  to  use. 

Example  1.     As  an  illustration,  consider  the  locus  of  the 
equation 

^+-!^+^=i-  (1) 

a^     1/      &  ^  ' 


246  ANALYTIC  GEOMETRY 

If  in  this  equation  z  be  set  equal  to  zero,  the  resulting  equa- 
tion represents  the  part  of  the  locus  which  lies  in  the  plane 
2;  =  0,  i.e.  in  the  ic^Z-pl^-ne. 

is  the  equation  of  the  intersection  of  the  locus  of  eq.  (1)  and 
the  a72/-plane. 

This  intersection  is  called  the  trace  of  eq.  (1)  in  the  cciz-plane. 
It  is  an  ellipse  with  semi-axes  a  and  h  lying  on  the  axes  of  x 
and  2/,  and  with  center  at  the  origin. 

Likewise  the  equations  of  the  locus  in  the  xz-  and  the  2/2;-planes 
are  shown  to  be  respectively  the  ellipses 

To  find  the  trace  of  the  locus  of  eq.  (1)  in  a  plane  parallel  to 
the  2/2;-plane  let  x  be  held  constant  in  eq.  (1)  and  y  and  z  be 
allowed  to  vary.  Letting  x  =  li.  in  eq.  (1),  the  resulting  equa- 
tion is 

1.2 

in  which  the  constant  term  —  is  transposed  to  the  right  side  of 

the  equation. 

This  equation  may  be  written 

.s>.  2 


This  is  the  equation  of  an  ellipse,  if  li^  <  a^,  with  axes  in  the 
planes  of  xy  and  xz,  the  values  of  the  semi-axes  being 


^,^6Va'-fc^^^^^,^cV«'-fc', 


LOCI  AND  THEIR  EQUATIONS 


247 


Hence  any  section  of  the  locus  of  eq.  (1)  by  a  plane  paralle  tctiie 
yz-plsine,  and  distant  less  than  a  from  the  origin  is  an  ellipse  with 
axes  in  the  planes  of  xy 
and  xz.  As  k  changes 
gradually  from  0  to  a, 
the  semi-axes  of  the 
ellipse  change  gradually 
from  b  and  c  to  0.  The 
locus  of  eq.  (1)  may  then 
be  thought  of  as  gener- 
ated by  an  ellipse  of 
gradually  varying  di- 
mensions moving  with 
its  axes  in  the  planes 
of  xy  and  xz.     The  locus  is  therefore   a  surface. 

Since  all  sections  parallel  to  three  mutually  perpendicular 
planes  are  ellipses,  the  figure  is  called  an  ellipsoid  (Fig.  161.) 

Example  2.     To  find  the  locus  of 

x^-{-2y^  =  Az.  (2) 

If  z  is  held  constant,  z  =  k,  the  equation  may  be  written 


Fig.  161. 


4:k     2k 


=  1, 


which  is  the  equation  of  an  ellipse  if  A;  >  0,  but  has  no  locus  if 
fc  <  0.  When  A;  =  0,  the  equation  is  satisfied  only  by  the  point 
(0,  0).  Therefore  a  section  of  the  locus  of  eq.  (2)  by  a  plane 
parallel  to  the  a;?/-plane  is  an  ellipse  if  the  plane  is  above  the 
fljy-plane,  but  there  are  no  points  below  the  a^^z-plane  which 
satisfy  the  equation. 

If  x=  0,  eq.  (2)  reduces  to  y^  =  2  z,  which  is  the  equation  of 
a  parabola  in  the  2/2-plane. 

If  y  =  0,  eq.  (2)  reduces  to  a;^  =  4  z,  which  is  the  equation  of 
a  parabola  in  the  iC2-plane. 

The  locus  of  eq.  (2)  may  therefore  be -thought  of  as  a  surface 


248 


ANALYTIC  GEOMETRY 


generated  by  an  ellipse,  moving  in  a  plane  parallel  to  the  xy- 
plane,  its  center  on  the  2;-axis,  and  so  changing  in  size  that  the 

ends  of  its  axes  are  always 
Z     .  on  the  curves  y^  =  2z  and 

x'  =  4:z.     (Fig.  162.) 

The  figure   is    called   an 
elliptic  paraboloid. 

217.  Locus  of  an  equation 
in  three  variables.  In  gen- 
eral an  equation  in  three 
variables  represents  a  sur- 
face. For  if  any  one  of  the 
variables  be  held  constant, 
an  equation  between  the 
other  two  variables  is  ob- 
tained, which  in  general  represents  a  curve,  as  was  found  in  the 
study  of  loci  in  two  variables.  The  locus  of  the  equation  in 
three  variables  is  then  such  that  in  general  its  intersections  by 
planes  parallel  to  the  coordinate  planes  are  curves.  Therefore 
the  locus  of  the  equation  is  in  general  a  surface. 


Fig.  162. 


EXERCISE  XLIX 

Discuss  and  sketch  the  loci  of  the  following  equations : 
1.    a;2  +  y2  4.  ^^2  =  ^2. 
%  y'^'  =  x  +  z. 

3.  x  +  y  -\-  z  =  l. 

4.  a;2  ^  y'2  +  4;^2  ^  1. 

5.  ic2  +  ?/2  _  ;52  _  0. 

6.  a;2  + 4^2  =  ^2. 

7.  a;  +  y  =  sin  «, 


CHAPTER  XIX 

THE  PLANE  AND  THE  STRAIGHT  LINE 

L     THE  PLANE 

218.  The  normal  equation  of  the  plane.  Let  p  be  the  length 
of  the  perpendicular  from  the  origin  upon  a  plane,  and  let  the 
direction  angles  of  this  perpendicular  be  a,  y8,  y. 


io,o,c) 

H             N^^ 

vX^P     ^ 

'^'y^^^"^"""'^ 

M 

\.      X 

-^               N 

\/^ 

^^^(a,o,o) 

,b,o) 

Fig.  163. 

Let  P  (x,  y,  z)  be  any  point  in  the  plane.  Project  the 
line  OP  and  also  the  broken  line  OM  -\-  MN -\-  NP  upon  the 
perpendicular.  (Fig.  163.)  These  projections  are  equal. 
(Art.  211.) 

.*.  a;  cos  a  +  ?/  cos  /8  +  2  cos  y  =  proj.  of  OP  on  OH,  (Art.  211) 
or     a?  cos  a  +  2/  cos  p  +  2!  cos  y  =P- 

Since  this  is  true  for  any  point  in  the  plane,  and  for  no 
other  points,  it  is  the  equation  of  the  plane. 

249 


250  ANALYTIC  GEOMETRY 

It  is  known  as  the  normal  equation  of  the  plane. 

219.  The  intercept  equation  of  the  plane.     If  the  above 
plane  meets  the  axes  in  {a,  0,  0),  (0,  b,  0),  and  (0,  0,  c),  then 

cos  a  = -^ ,  cos  i8  =  ^,  cos  y  =  ^ . 
a  be 

Substitute  these  values  of  cos  a,  cos  p,  cos  y  in  the  equation 

of  the  plane  and  there  results 

a     b     c 

220.  The  general  equation  of  the  first  degree  in  x,  y,  and  z. 

The  general  equation  of  first  degree  in  x,  y,  and  z  is 

Ax  +  By  +  Cz  +  I)  =  0.  (1) 

Consider  the  point  Q  whose  coordinates  are  the  coefficients 
of  X,  y,  and  z ;  i.e.  the  point  {A,  B,  C).  (Fig.  164.)  Let  OQ 
have  direction  angles  a,  fi,  y.     Then 


Fig.  164. 


COS«  =  A,cOS^=^,COBy  =  ;^, 


where 


OQ  =  -VA'  +  B'+0'. 


Dividing  eq.  (1)  through  by  ±  V^'  +  -B^  +  C%  it  may  be 


THE  PLANE  AND  THE  STRAIGHT  LINE        251 


written  in  the  form 


±  -^A^  +  B'  +  C'         ±  ^A^  +  B'+G'         ±  V^'  +  ^'  +  C^ 

-D 


±  VA"  +  B'  +  C 


(2) 


Let  the  sign  of  the  radical  be  chosen  so  that 


±  V^'  +  ^'+O^ 


is  positive,  and  let  p  = = .    Eq.  C2)  may  then 

be  written 

X  cos  a'  -\-  y  cos  P'  -\-  z  cos  y'  =  i?, 

in  which  a',  ^',  y',  are  the  same  as  a,  fi,  y,  or  are  the  direction 
angles  of  the  line  from  the  origin  to  (—  ^,  —  B,  —  C),  accord- 
ing as  the  positive  or  negative  sign  of  the  radical  is  chosen. 

In  either  case  the  equation  is  the  equation  of  a  plane  by 
Art.  218.     Therefore  the  equation 

Ax-\-  By  +  Cz  +  D  =  0, 

in  which  A,  B,  C,  D,  are  real  quantities,  is  the  equation  of  a 
plane.  If  p  is  the  length  of  the  perpendicular  from  the  origin 
to  the  plane,  and  a,  ft,  y,  are  the  direction  angles  of  this  per- 
pendicular, then 

cos  a  = cos  p  = 


±  \/A-'  +  B'  +  C-'  ±  VA^  +  B^  +  €'^ 

COS  7  = ■  ^  P  = .  the  same  sign 

of  the  radical  being  used  throughout,  and  so  chosen  that  p  is 
positive. 

221.   Distance  from  a  point  to  a  plane. 

(The  case  where  the  point  and  the  origin  are  on  opposite 
sides  of  the  plane  is  the  only  one  discussed  here.) 


252 


ANALYTIC  GEOMETRY 


Let  d  be  the  distance  from  (x^,  y^,  Zi)  to  the  plane  whose 
equation  is 

X  cos  a  +  y  cos  (3-\-z  cos  y=p. 

Through  (x^  yi,  z^  draw  a  plane  parallel  to  the  given  plane. 

Since  the  perpendicular  from  the 
origin  to  this  plane  is  in  the  same 
•(a;i,?/i,2i)  direction  as  that  from  the  origin 
to  the  given  plane,  the  equation  of 
the  second  plane  is 

X  cos  a  +  2/  cos  P-\-z  cos  y  =  p\ 
Since  (iCj,  2/1,  ^i)  is  on  this  plane, 
«i  cos  a  +  2/1  cos  /8  +  2^1  cos  y  =p'. 
But  d=p' —p. 

.•.  c?  =  a?i  cos  a  +  2/1  cos  p  +  «!  cos  y—p. 

The  student  should  show  that  if  the  point  and  the  origin 
are  on  the  same  side  of  the  plane,  the  above  formula  gives  the 
negative  of  the  distance  from  the  point  to  the  plane. 

From  the  above  it  follows  that  the  distance  from  (xu  2/1,  ^i) 
to  the  plane  Ax  -\-  By  -\-  Cz  -\-  D  =  0 


Fig.  165. 


is 


d  = 


Aoci  +  By  I  -\-Czi  +  jy 


222.  The  angle  between  two  planes.  Since  the  angle  be- 
tween two  planes  is  equal  to  the  angle  between  the  normals  to 
the  planes,  it  follows  that  the  angle  between  the  two  planes 

X  cos  «!  +  2/  cos  fti-hz  cos  yi  =Pi, 
and  X  cos  % H- 2/  cos  Pi-{-z  cos  y,  =  P2 

is  given  by 

cos  6  =  cos  «!  cos  ttg  4-  cos  /81  cos  ^2  +  cos  yi  cos  y2, 
and  that  the  angle  between  the  two  planes 

A,x-\-B,y+C,z  +  D,  =  Oy 
and  ^2»  +  ^22/+C'22  +  i>2  =  0 


THE'  PLANE  AND  THE  STRAIGHT  LINE        253 


is  given  by 

cos  6  =  ± 


A1A2  +  B1B2  +  C1C2 


y/A{'  +  B{'  +  Ci^  V^22  +  B2'  +  C22 


EXERCISE  L 

Find  the  lengths  and  direction  cosines  of  the  perpendiculars  from  the 
origin  upon  each  of  the  following  planes.  Reduce  each  equation  to  the 
normal  form. 

1.    2x-3i/  +  40  =  6.  2.   3x-6y-2z  =  0. 

3.    Sx-\-4:y=:2.  4.   x  +  y-{-z  =  l. 

Find  the  distance  from  the  following  points  to  the  planes : 

5.  From  (3,  1,  2)  to  2x-3?/  +  7;s  =  2. 

6.  From  (-1,  3,  2)  tox  +  2y-0  =  5. 

7.  From  (0,  0,  1)  to  2  x  -  y  =  4. 

8.  Find  the  angle  between  the  two  planes  of  example  1  and  example  2. 

9.  Find  the  angle  between  the  two  planes  of  example  3  and  example  4. 

II.     THE   STRAIGHT  LINE 

223.   The  equations  of  a  straight  line  through  two  points. 

Let  the  two  given  points  be  Pi(xi,  y^,  z^)  and  1*2(^2}  2/2>  ^2)-  I^6t 
P(x,  y,  z)  be  any  point  on  the 
line  through  Pj  and  P^.  Pro- 
ject PiP  and  P1P2  upon  the 
a>axis.  Then,  by  plane  geom- 
etry, 

M^M  ^  PiP 

M^M^     P1P2' 

X-Xi  ^PiP 

X2  —  Xi        Xir2 

In  like  manner  it  is  shown  that 

and  ^_^  =  AP. 

Z2-Z^        P1P2 

...    a?  -  xi  _  y  -  yi  _  g  -  jgi 

"      X2-OC1        2/2-2/1        «2  -  Zl 


Fig.  166. 


(1) 


254  ANALYTIC  GEOMETRY 

These  equations  are  therefore  the  equations  of  the  straight 
line. 

224.   The  equations  of  a  straight  line  through  a  given  point 

and  with  given  direction  cosines.     In  the  preceding  article  if 

the  line  makes  angles  a,  y8,  y  with  the  axes,  and  ii  d  =  P1P2, 

then 

3^2  —  ^  o     V2  —  V1  ^2  —  ^1 

cosa  =  ^^ -,   cos3  =  — — ^,   cos  7  =  -^^ -• 

d  d  d 

Substituting  the  values  of  ajg  — ajj,  1)2  — Vn  2:2  — 2:1  obtained  from 
these  equations  in  the  equation 

X2-Xy_     2/2-2/1     ^2-Zi 
there  results,  on  dividing  through  by  d, 

cos  a        cos  p       cos  "Y 

Hence  these   are   the   equations  of  a  straight  line  through 
(^1)  2/i>  z^  with  direction  angles  a,  /3,  y. 
Any  equations  of  the  form 

ag-a^i  _  y-Vi  _  z-zi 
I  m,  n 

are  the  equations  of  a  straight  line  through  (01^,  2/ij  ^^i)  with  di- 
rection cosines  proportional  to  /,  m,  n.  For  these  equations 
have  only  to  be  multiplied  by  V/^  -\-m^-\-n^  to  bring  them 
into  the  form 

x  —  Xx        _         y  —  Vx         _         Z—Zx 
I  m  n 


which  are  the  same  as  eqs.  (2),  since  the  denominators  in 
these  equations  are  the  direction  cosines  of  a  straight  line. 
(Art.  206.) 

225.   The  general  equations  of    a  straight  line.     Since  a 
straight  line  is  the  intersection  of  two  planes,  the  equations 


THE  PLANE  AND  THE  STRAIGHT  LINE        255 


of  two   planes  may  be  taken  as  the  general  equations  of   a 
straight  line.     Thus 

and  A<p;  +  JB22/  +  Og^  +  A  =  0, 

are  the  equations  of  a  straight  line. 

Since  one  straight  line  is  the  intersection  of  an  indefinite 
number  of  pairs  of  planes, 
the  same  straight  line  may- 
correspond  to  an  indefinite 
number  of  pairs  of  equa- 
tions of  first  degree. 

A  line  not  perpendicular 
to  the  a;-axis  may  be  repre- 
sented by  equations  of  the 
form 

y  =  mx  +  b, 
and    z  =  nx-{-c.  (Fig.  167.) 

If  it  is  perpendicular  to 
the  ic-axis,  but  not  to  the  2/-axis,  its  equations  may  be  written 

z  =  my-^b.     (Fig.  168.) 

If  it  is  perpendicular  to  both 

the  X-  and  iz-axes,  i.e.  is  parallel 

to  the  g-axis,  its  equations  may 

be  written 

x  =  a, 
y  =  b. 

EXERCISE  LI 
1.   Find  the  direction  cosines  of 


Fig.  168. 


the  line 


5 


-3_s+l 


256  ANALYTIC  GEOMETRY 

2.  Find  the  direction  cosines  of  the  line 

y  =  Sx  +  5, 

z  =  2x  +  l. 

3.  Prove  that  the  direction  cosines  of  the  line 

y  =  mx-{-b, 
z  =  nx  -\-  c 
are  proportional  to  1,  m,  n. 

4.  Prove  that  the  line 

_^  _  y-yi  _  g-gj 


I  m  n 

is  perpendicular  to  the  plane 

lx  +  my  +nz-\-p  =  0. 
5.   Find  the  angle  between  the  line 

x  —  S _y  _z  —  1 
2     ~4~  -  1 

and  the  perpendicular  to  the  plane 

3ic-22/  +  4^  =  0. 

-  6.   Find  the  angle  between  the  lines 

y-2_z-Z 
""-T-     3 

and  ^±i  =  ^.ll^^^  +  2. 

3-2 

7.  Find  the  angle  between  the  lines 

3x-2?/  =  4, 
4?/-20  =  l; 
and  a;  =  2y  +  3  =  40— 1. 

8.  Find  the  equations  of  the  line  through  (1,  —  1,  2)  which  makes 
equal  angles  with  the  axes. 

9.  Find  the  equations  of  a  line  through  (3,  4,  1)  and  (—2,  1,  3). 

10.  Find  the  equations  of  a  line  through  (3,  1,  -  2)  perpendicular  to 
the  plane  2aj— 3?/  +  42  =  0. 

11.  Find  the  equation  of  a  plane  through  (2,  1,  3)  parallel  to  the  line 
jc  =  2i/  +  4  =  3  0  —  1.  Also  the  equation  of  a  plane  perpendicular  to  the 
given  line  and  passing  through  the  given  point. 


CHAPTER   XX 

THE   QUADRIC   SURFACES 

226.    Definition.     The  quadric  surfaces,  or  conicoids,  are 

surfaces  whose  equations  are  of  the  second  degree  in  rectangu- 
lar coordinates  of  space. 

Certain  standard  forms  of  equations  of  second  degree,  formed 
by  analogy  to  the  standard  equations  of  second  degree  in  two 
variables,  will  be  studied  in  the  succeeding  articles. 


227.   The  ellipsoid. 


This  equation  has  already  been  discussed  in  Art.  216.     Only 
the  figure  is  shown  here. 


Fig.  169. 

If  two  of  the  quantities,  a,  6,  c,  are  equal,  e.g.  if  6  =  c,  the 
equation  reduces  to  that  of  the  spheroid, 

^'  +  ^  +  ^  =  1,    prolate  if  6<a,   oblate  if  5>a, 
8  267 


258  ANALYTIC  GEOMETRY 

If  a  =  6  =  c,  the  equation  reduces  to  that  of  the  sphere 
a?2  +  2/^  +  «^  =  a^- 

228.  The  hyperboloid  of  one  sheet. 

a^     h^     <f'       ' 

Sections  of  the  surface  represented  by  this  equation  by  planes 
parallel  to  the  a?2/-plane  are  of  the  form 

£f  4-^-1+!! 


^^  '   2/^2    ,    ^2\  +  T2  ^'2    .     ..'n  —  1- 


If  z  is  held  constant,  this  is  the  equation  of  an  ellipse. 
Sections  parallel  to  the  iC2!-plane  are  of  the  form 

^  _  2^  _  b^  —  y^ 

W  b' 

If  y  is  held  constant,  this  is  the  equation  of  an  hyperbola  with 
major  axis  parallel  to  the  ic-axis  if  y^  <,  b^,  and  with  major  axis 
parallel  to  the  2;-axis  if  y^  >  b^. 

For  y  =  b,ov  y  =  —b,  the  equation  represents  two  intersect- 
ing straight  lines 

-  +  -  =  0,  and--?  =  0. 
a     c  a     c 

The  hyperboloid  of  one  sheet  is  sketched  in  Eig.  170,  and 
a  few  sections  parallel  to  the  ic^-plane  are  indicated. 

Sections  parallel  to  the  yz-plane  are  also  hyperbolas,  and 
have  their  major  axes  parallel  to  the  y-  or  2;-axis  according  as 


THE  QUADRIC  SURFACES 


259 


the  distance  of  the  section  from  the  origin  is  less  than  oi 
greater  than  a.     The  two  sections  parallel  to  the  2/2;-plane  at 
the  distance  a  from  the  origin 
are  each   the   pair   of   straight 
lines 

2  +  ?  =  0,  and  2^-?=0. 
be  bo 

229.  The  hyperboloid  of  two 
sheets. 


a2       ^2      c2 


1. 


Any  section  of  the  surface 
parallel  to  the  a^^z-plane  is  an 
hyperbola  with  major  axes  par- 
allel to  the  a.'-axis,  the  major 
axis  and  conjugate  axis  both  in- 
creasing as  the  distance  of  the 
cutting  plane  from  the  «?/-plane 
increases,  but  their  ratio  re- 
maining equal  to  -• 
b 

A  like  remark  applies  to  sec-  yig.  170. 

tions  of  the   surface   made  by 
planes  parallel  to  the  a;2;-plane,  the  major  axis  being  parallel  to 

the  a^axis  and  the  ratio  of  the  axes  being  equal  to  -• 

c 

Sections  of  the  surface  parallel  to  the  yz--pla,n.G  are  of  the 

form 

^  ,z^  _a^     1 

b-     c^     a^ 


or 


+ 


2 


=  1. 


This  is  the  equation  of  an  ellipse  if  a;^  ->^  ^2^  )^^^  there  is  no 


260 


ANALYTIC   GEOMETRY 


locus  if  a^  <  a^.     When  a;  =  ±  a,  the  locus  is  a  point  (a,  0,  0), 
or  (-  a,  0,  0). 

Since  this  hyperboloid  consists  of  two  separate  parts,  it  is 
called  the  hyperboloid  of  two  sheets.  Only  one  part  is  shown 
in  the  figure.  The  other  part  is  symmetric  to  the  part  that  is 
shown  with  respect  to  the  yz-iglsme.     (Fig.  171.) 


Fig.  171. 

230.  The  elliptic  paraboloid. 

The  trace  in  the  fl??/-plane  is  the  origin  — -|-^  =  0.  The 
trace  in  the  a;2;-plane  is  the  parabola  x^  =  —  z.  The  trace  in  the 
2/2;-plane  is  the  parabola  y^=  —z.  Sections  of  the  surface 
parallel  to  the  a;?/-plane  are  of  the  form 

and  are  therefore  ellipses  if  h  and  c  are  of  the  same  sign,  but 
there  is  no  locus  if  k  and  c  are  of  opposite  sign.     Sections  of 


THE  QUADRIC  SURFACES 
the- surface  parallel  to  the  a^^i-plane  are  the  parabolas 


261 


k' 


with  vertices  at 


,  and  axes  parallel  to  the  2;-axis. 


a' 

[o,  K  f 
Sections  of  the  surface  parallel  to  the  i/^-plane  are  the  parabolas 

ckn 


with  vertices  at 


[M,f 


,  and  axes  parallel  to  the  2;-axis. 


The  locus  is  sketched  in  Fig.  172,  for  c  positive. 

Z 


i  Fig.  172. 

I 

231.    The  hyperbolic  paraboloid. 

^  _  ^  —  ? 

The  trace  in  the  a^y-plane  is  the  pair  of  straight  lines 

a      b  ah 

The  trace  in  the  a^^^-plane  is  the  parabola  x^  =i  —  z.     The  trace 

c 

in  the  i/^-plane  is  the  parabola  y^  = z.     Sections  parallel  to 


262 


ANALYTIC  GEOMETRY 


the  iC2/-plane  are  the  hyperbolas 

Sections  parallel  to  the  xz-iglane  are  the  parabolas 

s^_z      k^ 

Sections  parallel  to  the  2/0-plane  are  the  parabolas 

l^_z     If 
W         c      a'' 

The  locus  is  sketched  in  Fig.  173  for  c  positive. 


Fig.  173. 


232.  The  cone. 


a?2  ,  y^     «2 


r.+ 


6^     c2 


=  0. 


THE  QUADRIC  SURFACES 


263 


Let  this  surface  be  cut  by  the  plane  y  =  x  tan  $.  Let  »'  be 
the  distance  from  the  2-axis  to  any  point  P  on  the  intersection 
of  surface  and  plane.     Then 

y  —  x^  sin  B,    a;  =  «'  cos  0,  (Fig.  174), 


and 


or 


a;^^cos^^     a;'^sin^l9 

a?       "^       h^ 
62cos^+ a^sin^^    ,, 


^  =  0, 


-  =  0. 


Fig.  174. 


This  is  the  equation  of  the  intersection  of  the  plane  and  sur- 
face referred  to  rectangular  coordinates  in  the  cutting  plane. 
The  equation  can  be  factored  into  two  real  factors  of  first 
degree  in  a;'  and  z,  and  is  therefore  the  equation  of  two  straight 
lines.  Since  ic'  =  0  and  2  =  0  reduce  both  of  the  factors  to 
zero,  the  two  lines  pass  through  the  origin. 

Hence  any  plane  containing  the  2;-axis  intersects  the  surface 
in  two  straight  lines  through  the  origin. 

Moreover  any  plane  parallel  to  the  a;?/-plane,  2  =  A;,  intersects 
the  surface  in  the  curve 


an  ellipse. 


264 


A.NALYTIC  GEOMETRY 


Hence  the  locus  of 


ic^      y^      z^  _ 


-2  + 


--  =  0 


6^      (? 
is  a  cone  with  vertex  at  (0,  0,  0),  and 
with  the  section 

at  the  distance  c  from  the  a;?/-plane. 
(Fig.  175.) 

233.   The  right  circular  cone.     In 

the  equation  of  the  preceding  article 
if  a  =  b,  the  cone  becomes  a  right 
circular  cone. 

If  -  be  replaced  by  m,  the  equa 
c 

tion  of  the  right  circular  cone  be- 
FiG.  175.  comes 

If  x  =  0,  then  y^±  mz.  Therefore  the  straight  lines 
y  =  ±mz  are  the  intersections  of  the  cone  and  the  y2;-plane. 
Hence  the  quantity  m  is  the 
tangent  of  the  angle  between  an 
element  of  the  cone  and  its  axis, 
m  =  tan  if>,  in  Eig.  175. 


234.   The  conic  sections.      In 

the  equation  of  the  cone 
a:^  4-  2/^  —  mh^  =  0, 
let  the  y-  and  z-axes  be  rotated 
through  the  angle  $  to  the  new 
axes  OY'  and  OZ'.  The  old 
coordinates  y  and  z  of  any  point 
in  terms  of  the  new  coordinates 
y'  and  /  of  the  point  are  then 


Fig.  176, 


THE  QUADRIC  SURFACES  265 

given  by 

2  =  2'  cos  d  —  y^  sin  0, 
y  —  z^  sin  B  -\-y^  cos  Q. 

The  ^-coordinate  does  not  change.     (Fig.  176.) 

Substituting  in  the  equation  of  the  cone,  and  collecting  terms, 
the  equation  of  the  cone  referred  to  the  new  axes  is 

Q^  +  (sin2  ^  _  ^2  gQs2  0^  ^t2  ^  2  sin  ^  cos  ^  (1  +  w?)  y'z' 
+  (cos^  e-m''  sin2  6)  y'^  =  0. 

If  in  this  equation  y'  is  held  constant,  the  intersection  of  the 
cone  and  a  plane  parallel  to  the  a;2;'-plane  is  obtained.  Since 
the  X  and  z'  of  points  in  this  plane  are  the  same  as  their  pro- 
jections on  the  ajg'-plane,  the  equation  of  the  curve  of  intersec- 
tion is  of  the  form 

x'-{-az'^-hdz'+f=0, 
where  a  —  sin^  9  —  m^  cos^  6, 

c?  =  2  sin  ^  cos  ^ (1  +  m})y\ 

f  =(cos^e-7rv' sin' 0)y'^ 

A  discussion  of  this  equation  shows  that 

(1)  If  y'  =  0,  then  both  d  and  /  are  zero,  and  the  equation 
becomes 

x^-\-az'^  =  0. 

This  is  the  equation  of  a  point  if  a  >  0,  i.e.  if  tan^  6  >  m^;  of 
two  intersecting  lines  if  tan-  0  <  7n';  and  of  one  straight  line 
if  tan^  0  =  m^. 

(2)  y'^0. 

(a)  If  tan^  0  =  m^,  the  equation  is  of  the  form 

x'  +  dz'-\-f=0, 

which  is  the  equation  of  a  parabola. 

(&)  If  tan^  0  ^  m%  the  equation  is  of  the  form 

x'-\.az"-\-dz'+f=0, 


266 


ANALYTIC  GEOMETRY 


which  is  of  the  type  of  an  ellipse  or  hyperbola  according  as  a 
is  positive  or  negative,  i.e.  according  as  tan^  6  is  greater  than 
or  less  than  m^. 

If  ^  =  90°,  i.e. 
the  cutting  plane 
is  perpendicular  to 
the  axis  of  the 
cone,  the  equation 
reduces  to 

which  is  the  equa- 
tion of  a  circle. 

Hence,  if  a  right 
circular  cone  is  cut 
by  a  plane : 

(1)  passing 
through  the  vertex, 
the  intersection  is 
a  pair  of  lines,  a 
single  line,  or  a 
point,  according  as 
the  angle  which 
the  plane  makes 
with  the  axis  of 
the  cone  is  less 
than,  equal  to,  or 
greater  than  the 
angle  between  the 
axis  and  an  element 
of  the  cone ; 

Fig.  177.  (2)  ^^^*   passing 

through  the  ver- 
tex, the  intersection  is  an  hyperbola,  a  parabola,  or  an  ellipse, 
according  as  the  angle  between  the  plane  and  the  axis  of  the 


THE  QUADRIC  SURFACES  267 

cone  is  less  than,  equal  to,  or  greater  than  the  angle  between 
the  axis  and  an  element  of  the  cone. 

In  the  special  case  where  the  plane  is  perpendicular  to  the 
axis  of  the  cone,  the  intersection  is  a  circle.     (See  Fig.  177.) 

EXERCISE  Ln 

Describe  and  sketch  the  loci  of  the  following  equations  : 
1.   x^  +  y^  +  ^  z^  =  4:.  2.   x2  +  y2  _  4  ^^2  _  4. 

3.   z^-\-y'^  =  4:X.  4.   a;2-4(y2  +  2;2)  =0. 

5.  x^-\-2z^  =  y.  6.   z-x^  =  y^. 

y2  «2  3.2 

7.   ^  — -  — s  =  !•  S.  pv  =  Bt,  B  constant ;  p,  v,  t,  variables. 

62     c2      a2 

9.  ^4-?l  =  by.  10.   x^-z^  =  2y. 

a^     c2 

11.  (X  -  1)2 +(y  + 2)2  +(0  +  1)2  =16. 

12.  x^  +  y^  +  z^  =  ai      13.  x^  +  y^  +  z^  =  a^. 


CHAPTER  XXI 


SPACE  CURVES 

235.  Introduction.  In  this  chapter  a  few  curves  in  space, 
which  do  not  lie  in  a  plane,  will  be  considered,  and  the  equations 
derived. 

236.  The  helix.     The  helix  is  a  curve  traced  on  the  surface 

of  a  right  circular  cylinder 
by  a  point  which  advancies 
in  the  direction  of  the  axis 
of  the  cylinder  at  the  same 
time  that  it  rotates  around 
the  axis,  the  amount  of  ad- 
vance being  proportional  to 
the  angle  of  rotation. 

To  find  the  equations  of 
the  helix,  let  the  axis  of  % 
be  the  axis  of  the  cylinder 
on  which  the  helix  is  traced, 
a  the  radius  of  the  cylinder, 
h  the  amount  of  advance 
along  the  axis  to  each  radian 
of  rotation,  and  let  the  a;-axis 
be  chosen  to  pass  through 
a  point  of  the  helix.  Then, 
if  Q  is  the  angle  of  rotation 
around  the  axis,  the  values 
of  a;,  2/,  and  z  of  any  point  on 
Fig.  178.  the  curve  are 

x=  a  cos  ^, 

y  =  a  sin  ^, 

z^hB,  (Fig.  178.) 

268 


SPACE  CURVES 


269 


237.   The  curve  of  intersection  of  two  cylinders  of  unequal 
radii,  with  axes  intersecting  at  right  angles. 

Let  the  axes  of  the  cylinders  be  the  x-  and  ^/-axes  respectively, 
the    radii   a  and    h    respectively. 
(Fig.  179.)     The  equations  of  the 
surfaces  are  then 


and 


f  +  z'  =  a\ 

0?  +  Z^=z  h\ 


These  equations,  regarded  as  simul- 
taneous equations,  are  therefore 
the  equations  of  the  curves  of  in- 
tersection. 

The  equations  of  the  curve  may 
be  written  in  the  parametric  form,  / 
as  in  the  case  of  the  helix,  by  let- 
ting z  equal  some  arbitrary  function  of  another  variable  and 
then  solving  the  equations  for  x  and  y.     E.g.  if 


Fig.  179. 


then 
and 


z  —  a  sm  6, 

y  =  ±  a  cos  0, 

x=  ±  V^^  —  a'  sin^  0. 


Or  z  itself  may  be  considered  the  parameter,  and  the  equations 
written  in  the  parametric  form 


x=  ±-Vb^.-  z\ 
2/  =  ±  Va^  —  z^^ 
z  =  z. 


238.  The  curve  of  intersection  of  a  sphere  and  circular 
cylinder. 

Let  the  sphere  have  its  center  at  the  origin  and  radius  a, 
and  let  the  cylinder  have  its  axis  parallel  to  the  x-axis,  cutting 
the  2;-axis  at  2;  =  c,  and  radius  h. 


270  ANALYTIC  GEOMETRY 

The  equation  of  the  sphere  and  cylinder  are  then  respectively 

y?  -\- if-  ■\-  z^  =  o?i 
and  /  +  (2;  -  cf  =  h\ 

These  equations,  regarded  as  simultaneous  equations,  are  there- 
fore the  equations  of  the  curve  of  intersection. 

The  student  should  sketch  the  figure. 

The  coordinate  z  may  conveniently  be  considered  the  inde- 
pendent variable  and  have  arbitrary  values  assigned  to  it,  the 
corresponding  values  of  x  and  y  being  then  computed  from  the 
equations.  Corresponding  to  one  value  of  z  four  points  are 
obtained,  in  general. 

Exercise.  Letting  a  =  5,  6  =  2,  c  =  3,  find  four  points  on  the  curve 
corresponding  to  ^  =  2.  How  many  points  of  the  curve  are  there  having 
z=\^     How  many  having  2  =  5  ? 

239.  General  equations  of  a  space  curve.  If  the  equations 
of  two  surfaces  are  known,  these  equations,  regarded  as  simul- 
taneous equations,  are  satisfied  by  all  points  common  to  the 
two  surfaces,  and  by  only  those  points.  The  equations  of  the 
two  surfaces  are  therefore  together  the  equations  of  the  curve 
of  the  intersection  of  the  two  surfaces. 

EXERCISE  LIII 

1.  A  screw  has  8  threads  to  the  inch.  The  diameter  of  the  screw  is  \ 
inch.     What  are  the  equations  of  the  edge  of  the  threads  ? 

2.  A  point  starts  at  the  base  of  a  right  circular  cone  and  traces  a  curve 
on  the  surface,  advancing  in  the  direction  of  the  axis  of  the  cone  propor- 
tional to  the  angle  of  rotation  about  the  axis.  Find  the  equations  of 
the  curve. 

3.  Similar  to  example  2,  using  a  hemisphere  instead  of  a  cone. 

4.  Find  the  polar  equation  of  the  projection  of  the  curve  of  example  2 
upon  the  plane  of  the  base  of  the  cone.    Trace  the  curve  of  projection. 

5.  Find  the  polar  equation  of  the  projection  of  the  curve  of  example  3 
upon  the  plane  of  the  base  of  the  hemisphere. 


CHAPTER   XXII 

TANGENT  LINES  AND  PLANES 

240.  Introduction.  In  the  plane  a  knowledge  of  derivatives 
was  found  to  be  important  in  obtaining  the  equations  of  tangent 
lines  to  curves.  In  space,  also,  derivatives  play  an  important 
part  in  the  deduction  of  the  equations  of  tangent  planes  to  sur- 
faces and  of  tangent  lines  to  curves.  But  in  space  a  somewhat 
extended  conception  of  derivatives  is  necessary,  for  the  number 
of  variables  has  increased  from  two  to  three. 

241.  Partial  Derivatives.  Consider  an  equation  which  ex- 
presses 2  as  a  function  of  two  independent  variables,  x  and  y. 

E.g.  z  =  2x^^Zxy^  +  ^f.  (1) 

If  y  is  regarded  as  a  constant  and  the  derivative  of  z  taken  with 
respect  to  x,  the  result  is 

4.x^Sf. 
This  result  is  called  the  partial  derivative  of  z  with  respect 

to  Xj  and  is  denoted  by  the  symbol  — .     Thus, 

ax 

Similarly,  -—  =  6xy  +  15  y\ 

dy 

In  eq.  (1)  let  x  and  y  take  the  values  Xq  and  y^  respectively. 
Then  z  takes  a  corresponding  value,  Zq.     Then 

z^  =  2x^  +  3  Xoyo^  4-  5  2/o^ 

Let  X  take  an  increment.  Ax.  Then  z  takes  a  corresponding 
increment.  Let  this  increment,  which  is  due  to  the  change  in 
X  only,  be  denoted  by  A^z.     Then 

Zo  +  A,2  =  2(xq  +  Axy  +  S{xq  +  Ax)yJ^  +  5  y^^ 
271 


272  ANALYTIC  GEOMETRY 

From  the  definition  of  a  derivative,  it  follows  that  the  value 


of 


dz 
dx 


is  the  limiting  value  of  -^  as  Ax   approaches   zero. 

■  Ax 


The  student  can  easily  check  this  by  computing  the  value  of 
-^  from  the  above  equations  and  finding  the  limiting  value. 

i^X 

In  general,  if  f(x,  y),  read  "/  of  x  and  2/,"  is  used  to  denote 
any  function  of  x  and  2/,  then,  if 

^  =f(^,  y), 


the  values  of 


dz 


and 
If 

then 


dx 

dz\ 


dz 


are  defined  by 


and 

0.  ^0  oy 

=  iimif./(^o  +  Ax,  yp)  -f(xo,  yo) 
dx\x^,y^Ax  =  0  Ax 

x„  y,Ay  =  0  Ay 

u  =  F(x,y,  2), 


du 


is  defined  by 
dx  xo-yo'^o 

du  ^  i-jjj.^  Fjxp  +  Ax,  1/0,  gp)  -  F(xq,  yo,  gp) 

dx  x^,  y^,  Zf,     Ax  =  0  Ax 

and  similarly  for  the  other  partial  derivatives  of  u. 

EXERCISE  LIV 

1.  Find  the  partial  derivative  of  z  with  respect  to  x  and  y  for  the  values 
x  =  2,y  =  3,ilz  =  '6x^-  5xy^  +  2f. 

2.  In  the  equation  of  example  1,  letting  x  =  2,  i/  =  3,  compute  the 
increment  in  z  due  to  an  increment  of  .  1  in  x.  Also  the  increment  in  z 
due  to  an  increment  of  .1  in  y. 

3.  If  u  =  Sx^  -\-2xyz  -\- y^  +  5 x^z  +  yz^,  find  the  partial  derivatives  of 
u  with  respect  to  each  of  the  variables  x,y,z. 

4.  In  z  =  2x^  ^Sxy  +  y,  find  the  value  of  — ,  (1)  by  differentiating, 

dx. 

regarding  y  as  constant ;  (2)  by  giving  x  an  increment,  Ax,  computing 

A  z 
AgZ,  and  finding  the  limiting  value  of  —^ . 

Ax 


TANGENT  LINES  AND  PLANES 


273 


242.   The  tangent  plane  to  a  surface.     Let 

F{x,y,z)  =  () 
be  the  equation  of  a  surface. '   Let  P(iCo5  2/oj  ^^o)  be  any  point  on 
this  surface,  and  let  the  surface  be  cut  by  a  plane  parallel  to 
the  2;-axis  and  passing  through  P.     (Fig.  180.)     The  equation 
of  such  a  plane  is 

y  =  mx  +  h. 


Fig.  180. 

Let  P\xq  +  \x,  yo-\-Ay,  Zq  +  A^:)  be  any  other  point  on  the 
intersection  of  the  surface  and  plane.     Then 

yo  +  Ay  =  m(.ro  +  Ao;)  +  b, 
and  ?/o  =  7)iXq  +  b. 

.*.    Ay  =  mAxj 

Ay 

Ax 

The  equation  of  the  line  through  P  and  P'  is 

x-x^^y-y^^z-z^ 

Ax  Ay 

x-Xq     y-yo 


or 


or 


Ay 
Ax 


Az   ■ 

Zj-Z^ 

Az 
Ax 


(Art.  223) 


274  ANALYTIC  GEOMETRY 

Let  P'  approach  the  limiting  position  P.  The  line  through  P 
and  P'  approaches  the  limiting  position  of  tangency  at  P  to 
the  curve  of  intersection  of  the  plane  and  surface,  and  hence 
of  tangency  at  P  to  the  surface.  The  equation  of  this  tangent 
line  is  therefore 

1  711  n 

Az 
where  n  is  the  limiting  value  of  — ^  as  P'  approaches  P. 

Ax 

To  find  the  value  of  n,  let  F(x,  y,  z)  be  represented  by  u ; 
u  =  F(x,y,z). 
Since  P'  and  P  are  both  on  the  surface,  therefore 

F(xq  +  Ax,  ?/o  +  Ay,  Zo  4-  Az)  =  0, 
and  F  {xq,  yo,  Zq)  =  0. 

^.^  F(xq  +  Ax,  yo  +  Ay,  Zp  +  Ag)  -  F{xq,  yp,  Zp)  _  ^ 
Ax 
This  equation  may  be  written 

Fjxp  +  Ax,  j/o  4-  Ay,  Zp  +  Az)  —  F(xq,  y^  +  Ay,  Zq  +  Az) 
Ax 
I  F{xp,  yp  +  Ay,  Zp  +  Az)  -  F(xp,  y^,  Zp  +  Az)  ^  Ay 
Ay  Ax 

,  F(xp,  yp,  Zp 4-  Az)  - Fjxp,  yp,  Zp)  ^  Az  ^^^ 
Az  Ace 

If  Ay  and  Az  were  held  constant,  and  Aa;  allowed  to  approach 
zero  as  a  limit,  the  first  term  of  this  equation  would  approach 
the  limiting  value 

du 

Since,  however,  as  P'  approaches  P,  Ax,  Ay,  and  Az  all 
approach  zero,  the  limiting  value  of  the  first  term  is 

du 

dx 


TANGENT  LINES  AND  PLANES 


275 


Likewise  the  second  and  third  terms  approach  the  limiting 
values 

du 


m 


and 


By 

du 
dx 


*0'  ^0'  *0 


*o'  2'o»  "o 


du  ,       du  ,      du 

\-m 1- w  — 

dx  dy  dz 


0, 


(2) 


the  values  of  the  partial  derivatives  being  taken  at  (xq,  y^,  z^. 

Equation  (2)  expresses  the  value  of  n  in  terms  of  m.  If 
eq.  (2)  were  solved  for  n,  and  the  value  substituted  in  eqs.  (1), 
the  equation  of  the  tangent  at  P  to  the  curve  of  intersection  of 
the  plane  and  surface  would  be  obtained.  The  elimination  of 
m  between  the  eqs.  (1)  would  then  result  in  an  equation  be- 
tween the  coordinates  of  points  on  any  tangent  line  to  the  sur- 
face at  P,  i.e.  the  equation  of  the  locus  of  all  tangent  lines  that 
can  be  drawn  to  the  surface  at  P. 

The  elimination  of  m  and  n  is  most  easily  affected  by  solving 
eqs.  (1)  for  m  and  n  and  substituting  their  values  in  eq.  (2). 
The  result  is 


/  \  du 

{x-x^)-- 

dx 


+  (z-Zq) 


'^O'  ^0'  ^0 


0. 


*0'  ^0'  *0 


Since  this  is  an  equation  of  first  degree  in  x,  y,  and  z,  it  is  the 
equation  of  a  plane. 

Hence  all  tangent  lines  to  a  surface  at  a  given  point  lie  in  a 
plane.  This  plane  is  called  the  tangent  plane  to  the  surface  at 
that  point. 


Since 


u  =  F{x,  y,  z), 


the  symbol  — —  may  be  used  instead  of  — 
dx  dx 


Hence,  if 


F{x,y,z)  =  0 


276 


ANALYTIC  GEOMETRY 


is  the  equation  of  any  surface,  then 

+  (2/-2/o)-T- 


+  (^-^o)f 


=  0 


*0>  3/0)  ^0 


is  the  equation  of  the  tangent  plane  to  the  surface  at  (xq,  2/0,  Zq) 
243.   Illustration.     Consider  the  ellipsoid, 


Here 

and 


H^,y.^)=^,-^ 


y 


+  ^— 1, 


h'  '  c" 


dF^2x    §F^2y    dF^2_z 

dx       a^  '   dy       b^  '   dz       (?  ' 


and  hence  the  equation  of  the  tangent  plane  at  (a^o,  2/0?  ^0)  is 


(^-«^o)^4-(y-2/o)^«H-(^-^o)^^  =  0. 


Since 


^2  -f-  52  -^  c2 


1, 


the  equation  of  the  tangent  plane  becomes 

a^'^  b^^  &        ' 

244.   The  normal  to  a  surface.     A   line  perpendicular  to  a 
tangent  plane  to  a  surface  at  the  point  of  tangency  is  called  a 
normal  to  the  surface  at  that  point. 
If  the  equation  of  the  surface  is 

F(x,y,z)=0, 
the  equation  of  the  tangent  plane  has  been  found  to  be 
dF 


(X-Xo) 


dx 


+  (2/-2/o)^- 

«o'  ^'o'  ^0  ^y 


"O-  ^0' 


=  0. 


"^0'  ^0'  ^0 


The  equations  of  a  line  perpendicular  to  this  plane  and  passing 
through  (xq,  2/0,  Zq)  are  therefore 


X—Xn 


dF 

dx 


'qi  Vq)  ^q 


y-yo 

w 


Z  —  Zq 


dF 

dz 


(Arts.  220,  224.) 


*0'  ''o'  "0 


TANGENT  LINES  AND  PLANES 


277 


If  the  equation  of  the  surface  is  given  in  the  form 

then  F(x,  y,z)=z-  f(x,  y), 

and      ^  =  -^  =  -^    M!  =  _3£  =  _^    Ml 
dx  dx  dx^    dy  dy  dy^    dz 

The  equations  of  the  normal  then  become 


1. 


y-Vo 


60 

dx 


"^0'  ^0'  ''o 


dz^ 
dy 


Z  —  Zq 

-1 


"O'  2'0'  ^0 


245.  The  tangent  line  to  a  space  curve.  Let  P(xo,  y^,  Zq)  and 
P'  (a^o  -f  Ax,  2/0  +  ^y,  ^0  +  ^^)  be  two  points  on  a  curve.  The 
equations  of  the  line  through  these  points  are 

^  —  ^0 y  —  yo ^  —  ^0 


Ax 


Ay 


Az 


(Art.  223.) 


p/^ 

A2 

r 

/Ay 

X 

Fia.  181. 
If  x,  y,  and  2;  are  functions  of  some  independent  variable,  t 
(compare  the  equations  of  the  helix,  Art.  236),  Ax,  Ay,  and  Az 
will  depend  upon  A^.     Let  the  above  equations  be  multiplied 
by  At.     Then 

x-Xo     y-yo     z  —  Zo 


Ax 
At 


Ay 
At 


Az 
At 


278  ANALYTIC  GEOMETRY 

As  P'  approaches  coincidence  with  P,  the  ratios  ^^^ 

Ai'  A«'  A« 

approach  the  values  of  the  derivatives  of  x,  y,  and  z  respectively 
at  {xq,  2/0,  z^.  The  line  through  P  and  P'  approaches  at  the 
same  time  the  limiting  position  as  tangent  to  the  curve  at  P. 
Hence  the  equations  of  the  tangent  to  the  curve  at  {xq,  y^^  z^  are 

x-X(,     y-yo     z-Zq 


dx 

dy 

dz 

—                             K^) 

dt 

#0        dt 

t,        dt 

<0 

If  the  equations  of  the  curve  are  the  simultaneous  equations 

two  surfaces, 

f(x,y,z)  =  0, 

<f>{x,y,z 

0  =  0, 

.. 

dx     dii     dz 
the  values  of  — >  —  >  —    may  be  obtained  as  follows:  Since 
dt     dt     dt        ^ 

P'  and  P  are  on  the  surface /(^j,  y,  z)=  0,  therefore 

/(xo  -+  A«,  ?/o  +  A?/,  Zq  +  ^z)  =  0, 
and  /(i»o,  2/0, 2;o)  =  0. 

.  /(a^o  +  Aar,  yo  +  Ay,  ^p  +  Ag)  -/(a?o,  yp,  gp)  ^  q 
Ai 

Treating  this  expression  as  was  done  in  Art.  242,  there  results 
dfdxdfdy      §fdz_r. 
dx  dt      dy  dt      dz  dt 

Similarly,  d±dx     d^dy      d^dz^^ 

^'  dxdt       dy  dt       dzdt        ' 

the   values   of  all  the  derivatives  being  taken  at  the  point 

From  these  equations  there  result 

^  (%  ^ 

dt  _  dt  _  dt 

^d^  _d_f^d^'~  dfd^_dfd^'~  d^d^_dfd^' 

dy  dz       dz  dy      dz  dx      dx  dz      dx  dy      dy  dx 


TANGENT  LINES  AND  PLANES  279 

Multiplying  the  members  of  eq.  (1)  by  the  corresponding 
members  of  this  equation,  there  result  as  the  equations  of 
the  tangent  line  at  (xq,  y^,  z^  to  the  curve  whose  equations  are 

f(x,y,z)  =  0, 
and  <f>  (x,  y,  z)  =  0, 

«  -  ^0  y-Po 


By  Bz 

_BfB_^\ 

~  fBf  B<t>  _  Bf  Bct>\ 
\Bz  Bx      BxBzJ,^,y^,,^ 

Z-Zo 

/BfBcl>_BfB<f>\ 
\BxBy      ByBxJ,^,y^,,^ 

246.   Illustrations.     Example  1.    To  find  the  equations  of 
the  tangent  to  the  helix  at  any  point. 
The  equations  of  the  helix  are 

x  =  a  cos  0, 
y  =  a  sin  0, 

z  =  be.  (Art.  236.) 

dx  .    ^ 

-^  =  a  cos  6. 
dO  ' 

—  —  h 

de~ 

Hence  the  equations  of  the  tangent  to  the  helix  at  a  point 
where  6  =  6^  are 

x  —  a cos  Op  _y  —  a  sin  Oq _z  —  bOp 
—  a  sin  6q  a  cos  $q  b 

Example  2.     To  find  the  equations  of  the  tangent  to  the 
curve  of  intersection  of  the  cylinders 

y^-\-z^  =  a% 
aP  +  z^r^bK 


280  ANALYTIC  GEOMETRY 

Let  f{x,y,z)  =  y''-\-z^~a% 

1  =  0,       g=2„     1  =  2, 

?.-'-    ?  =  «'      t  =  ^^- 
Therefore  the  equations  of  the  tangent  at  {xq,  yo,  Zq)  are 
x  —  Xq  ^  y  —  yo  ^  z  —  Zq  ^ 
2/02^0         ^6^0       —  i»o2/o 


EXERCISE  LV 

Find  the  equation  of  the  tangent  plane  to  each  of  the  fol- 
lowing ten  surfaces: 

1.    a;2  +  y-2  -\-  z'^  =  r^. 


3. 

a;2     ,/2     ^2 
a2     6-2     c2       ' 

5. 

X_2        y_2_£. 

a2^&2     c 

7. 

^+?-!-^!  =  o. 

2. 

a2     62  ^  c2 

4. 

x2+y2=2pX. 

6. 

a-.2     y2_ 
a2     62       • 

8. 

i}v  =  EL 

62 

9.   xyz  =  c.  10.   0  =  3  a; +  2!/. 

11.  Prove  that  the  direction  cosines  of  the  tangent  to  the  helix  are 
—  a  sin  00      a  cos  do  6 


vV  +  62      Va2  +  62     Va2  +  6^ 

(Note  that  the  angle  between  the  tangent  and  the  ^-axis  is  constant.) 

12.  If  the  point  generating  the  helix  advances  in  the  direction  of  the 
axis  of  the  cylinder  j^^  of  the  radius  of  the  cylinder  at  each  revolution, 
find  the  angle  between  the  tangent  to  the  helix  and  an  element  of  the 
cylinder. 

13.  Find  the  equations  of  the  tangent  to  the  helix  at  the  point  where 
d  =  30°. 

Find  the  equations  of  the  tangent  to  the  curve  of  intersection  of  each 
of  the  following  pairs  of  surfaces. 


TANGENT  LINES  AND  PLANES  281 

14.  1/2  +  ^2  =  1^   x^  +  2  y^  -\-  4:  z'^  =  4,  Bit  a,  point  where  z  =  \. 

15.  z  +  2  2/2  =  4,   x2  +  ?/2  _  2  =  0,  at  (1,  1,  2). 

16.  02  4.  2  2,2  =  4^   a;2  +  2/2  _  ^2  =  0,  at  (1,  1,  V2). 

17.  Prove  that  the  direction  cosines  of  the  normal  to 

F{x,  y,z)  =  0 
at  any  point  (x,  y,  z)  are 

dF  dF 

dx  By 


<m<fyy-m  m^m^m 


dF 


18.   Prove  that  the  direction  cosines  of  the  normal  to  the  surface 
z  =f(x,  y) 
at  any  point  {x,  2/,  z)  are 

dz                                   dz                                 _  J 
dx 5y ^ 

M%y^{%f  v-(ir-(ir'  v^w^^' 


TABLES 

TEIGONOMETRIC   FORMULAS 


sin^  A -\- cos^  A  =  1  sin  (  -  —  ^  )  =  cos^ 


sin  J. 

CSC  ^  =  1 

cos  J. 

sec^  =  l 

tan  ^  cot  ^  =  1 

tan  J.= 

sin^ 
cos^ 

(iotA  = 

cos^ 
sin^ 

sm(- 

-A)  = 

—  sin^ 

cos(- 

-A)  = 

cos  J. 

tan(- 

-A)  = 

—  tan  A 

cot(- 

-A)  = 

—  cot^ 

sec(- 

-A)  = 

sec^ 

csc(- 

-A)  = 

—  CSC  J. 

sec^^  — tan2^  =  l  cos(  ^— ^  j  =  sin^ 

csc^  J.  —  cot^  J.  =  1  tanf  ?  — ^  l  =  cot^ 


(1-")= 


sin  (  -  +  ^  J  =  COS  ^ 
cos  l-4-A]=  —  sin  A 


tan  (  -  +  ^^  =  —  cot  A 

sin  (tt  —  A)  =  sin  A 
cos  (tt—  A)  =  —  cos  A 
tan  (tt  —  A)  =  —  tsLuA 
sin  (ir-^  A)  =  —  sin  A 
cos  (tt  4-  J.)  =  —  cos  A 
tan  (Tr-\-A)=     tan  ^ 

sin  (^  +  2  mr)  =  sin  ^ 
cos  (^  +  2  wtt)  =  cos  ^ 
tan  (A -{-2  mr)  =  tan  A 
(n  a  pos.  or  neg.  integer) 


282 


TRIGONOMETRIC  FORMULAS 


283 


sin  {A±  B)  =  sin  A  cos  B  ±  cos  A  sin  B 
cos  (A±B)  =  cos  ^  cos  -B  T  sin  A  sin  B 
tan  ^  ±  tan  B 


t8i.Ji(A±B)  = 


1  q:  tan  A  tan  5 


sin  2  ^  =  2  sin  A  cos  ^ 


2sin2ft  =  l-cos^ 


cos  2  J.  =  cos^  ^  —  sin^  A 
=  l-2sin2J. 
=  2cos2J.-l 


2cos2  — =  l  +  cos^ 

2 

,      ^     1  —  cos  A 
tan  —  =  —, — — - 
2  sm  A 


tan  2  ^  = 


2  tan^ 


sin^ 


l-tan^^ 

sin  3  ^  =  3  sin  ^  —  4  sin^  ^ 
cos  3^  =  4  cos^  A  — 3  cos  ^ 

.      .  ,     .     T5      o   •    A-^B       A  —  B 

sin  ^  +  sm  ^  =  2  sin  — ^ —  cos  — - — 

2  2 

sin^  -  sin^  =  2  cos  ^  "^  ^  sin  i^—-^ 


1  +  cos  ^ 


A  ,         DO        A-hB       A  —  B 
cos  ^  -|-  cos  B  =  2  cos  — - —  cos 


2 


2 
A-B 


AA-B 

cos  ^  —  COS  J5  =  —  2  sin  — '^ —  sin 


A 

0° 

30° 

45° 

60° 

90° 

180° 

270° 

360° 

sin^ 

0 

1 

2 

V2 

2 

V3 

2 

1 

0 

-1 

0 

cos^ 

1 

V3 
2 

V2 
2 

1 
2 

0 

-1 
0 

0 

1 

tan^ 

0 

V3 
3 

1 

V3 

QO 

GO 

0 

284  ANALYTIC  GEOMETRY 

LOGARITHMS   OF   NUMBERS 


N 

O 

1 

2 

3 

4 

5 

6 

7 

8 

9 

lO 

0000 

0043 

0086 

0128 

0170 

0212 

0253 

0294 

0334 

0374 

11 

0414 

0453 

0492 

0531 

0569 

0607 

0645 

0682 

0719 

0766 

12 

0792 

0828 

0864 

0899 

0934 

0969 

1004 

1038 

1072 

1106 

13 

1139 

1173 

1206 

1239 

1271 

1303 

1335 

1367 

1399 

1430 

14 

1461 

1492 

1523 

1553 

1584 

1614 

1644 

1673 

1703 

1732 

15 

1761 

1790 

1818 

1847 

1875 

1903 

1931 

1959 

1987 

2014 

16 

2041 

2068 

2095 

2122 

2148 

2175 

2201 

2227 

2253 

2279 

17 

2304 

2330 

2355 

2380 

2405 

2430 

2455 

2480 

2604 

2529 

18 

2553 

2577 

2601 

2625 

2648 

2672 

2695 

2718 

2742 

2765 

19 

2788 

2810 

2833 

2856 

2878 

2900 

2923 

2945 

2967 

2989 

20 

3010 

3032 

3064 

3075 

3096 

3118 

3139 

3160 

3181 

3201 

21 

3222 

3243 

3263 

3284 

3304 

3324 

3345 

3365 

3386 

3404 

22 

3424 

3444 

3464 

3483 

3502 

3522 

3641 

3660 

3579 

3698 

23 

3617 

3636 

3655 

3674 

3692 

3711 

3729 

3747 

3766 

3784 

24 

3802 

3820 

3838 

3856 

3874 

3892 

3909 

3927 

3945 

3962 

25 

3979 

3997 

4014 

4031 

4048 

4065 

4082 

4099 

4116 

4133 

26 

4150 

4166 

4183 

4200 

4216 

4232 

4249 

4266 

4281 

4298 

27 

4314 

4330 

4346 

4362 

4378 

4393 

4409 

4425 

4440 

4456 

28 

4472 

4487 

4502 

4518 

4533 

4548 

4564 

4579 

4594 

4609 

29 

4624 

4639 

4654 

4669 

4683 

4698 

4713 

4728 

4742 

4757 

30 

4771 

4786 

4800 

4814 

4829 

4843 

4857 

4871 

4886 

4900 

31 

4914 

4928 

4942 

4955 

4969 

4983 

4997 

6011 

6024 

5038 

32 

5051 

5065 

5079 

5092 

6105 

5119 

5132 

5145 

5169 

5172 

33 

5185 

5198 

5211 

5224 

5237 

5250 

5263 

5276 

5289 

5302 

34 

5315 

5328 

5340 

5353 

5366 

5378 

5391 

5403 

5416 

5428 

35 

5441 

5453 

5465 

5478 

5490 

5502 

5514 

5527 

5639 

6551 

36 

5563 

5575 

5587 

5599 

5611 

5623 

5635 

5647 

5658 

5670 

37 

5682 

5694 

5705 

5717 

5729 

6740 

5752 

5763 

5775 

5786 

38 

5798 

5809 

5821 

5832 

5843 

5855 

5866 

6877 

5888 

6899 

39 

5911 

5922 

5933 

5944 

5955 

5966 

5977 

5988 

5999 

6010 

40 

6021 

6031 

6042 

6053 

6064 

6075 

6086 

6096 

6107 

6117 

41 

6128 

6138 

6149 

6160 

6170 

6180 

6191 

6201 

6212 

6222 

42 

6232 

6243 

6253 

6263 

6274 

6284 

6294 

6304 

6314 

6325 

43 

6335 

6345 

6355 

6365 

6375 

6385 

6396 

6406 

6415 

6425 

44 

6435 

6444 

6454 

6464 

6474 

6484 

6493 

6503 

6513 

6522 

45 

6532 

6542 

6551 

6561 

6571 

6580 

6590 

6699 

6609 

6618 

46 

6628 

6637 

6646 

6656 

6665 

6676 

6684 

6693 

6702 

6712 

47 

6721 

6730 

6739 

6749 

6758 

6767 

6776 

6785 

6794 

6803 

48 

6812 

6821 

6830 

6839 

6848 

6857 

6866 

6875 

6884 

6893 

49 

6902 

6911 

6920 

6928 

6937 

6946 

6955 

6964 

6972 

6981 

50 

6990 

6998 

7007 

7016 

7024 

7033 

7042 

7060 

7059 

7067 

51 

7076 

7084 

7093 

7101 

7110 

7118 

7126 

7135 

7143 

7152 

52 

7160 

7168 

7177 

7185 

7193 

7202 

7210 

7218 

7226 

7235 

j54 

7243 

7251 

7259 

7267 

7275 

7284 

7292 

7300 

7308 

7316 

7324 

7332 

7340 

7348 

7356 

7364 

7372 

7380 

7388 

7396 

LOGARITHMS  285 


N 

o 

1 

2 

3 

4 

5 

6 

7 

8 

9 

55 

7404 

7412 

7419 

7427 

7435 

7443 

7451 

7459 

7466 

7474 

56 

7482 

7490 

7497 

7505 

7513 

7620 

7528 

7536 

7643 

7551 

67 

7559 

7566 

7574 

7582 

7589 

7597 

7604 

7612 

7619 

7627 

58 

7634 

7642 

7649 

7657 

7664 

7672 

7679 

7686 

7694 

7701 

59 

7709 

7716 

7723 

7731 

7738 

7745 

7752 

7760 

7767 

7774 

60 

7782 

7789 

7796 

7803 

7810 

7818 

7825 

7832 

7839 

7846 

61 

7853 

7860 

7868 

7875 

7882 

7889 

7896 

7903 

7910 

7917 

62 

7924 

7931 

7938 

7945 

7952 

7959 

7966 

7973 

7980 

7987 

63 

7993 

8000 

8007 

8014 

8021 

8028 

8035 

8041 

8048 

8056 

64 

8062 

8069 

8075 

8082 

8089 

8096 

8102 

8109 

8116 

8122 

65 

8129 

8136 

8142 

8149 

8156 

8162 

8169 

8176 

8182 

8189 

66 

8195 

8202 

8209 

8215 

8222 

8228 

8235 

8241 

8248 

8254 

67 

8261 

8267 

8274 

8280 

8287 

8293 

8299 

8306 

8312 

8319 

68 

8325 

8331 

8338 

8344 

8351 

8357 

8363 

8370 

8376 

8382 

69 

8388 

8395 

8401 

8407 

8414 

8420 

8426 

8432 

8439 

8446 

70 

8451 

8457 

8463 

8470 

8476 

8482 

8488 

8494 

8500 

8506 

71 

8513 

8519 

8525 

8531 

8537 

8543 

8549 

8555 

8561 

8567 

72 

8573 

8579 

8585 

8591 

8597 

8603 

8609 

8615 

8621 

8627 

73 

8633 

8639 

8645 

8651 

8657 

8663 

8669 

8675 

8681 

8686 

74 

8692 

8698 

8704 

8710 

8716 

8722 

8727 

8733 

8739 

8746 

75 

8751 

8756 

8762 

8768 

8774 

8779 

8785 

8791 

8797 

8802 

76 

8808 

8814 

8820 

8825 

8831 

8837 

8842 

8848 

8854 

8859 

77 

8865 

8871 

8876 

8882 

8887 

8893 

8899 

8904 

8910 

8916 

78 

8921 

8927 

8932 

8938 

8943 

8949 

8954 

8960 

8965 

8971 

79 

8976 

8982 

8987 

8993 

8998 

9004 

9009 

9015 

9020 

9025 

80 

9031 

9036 

9042 

9047 

9053 

9058 

9063 

9069 

9074 

9079 

81 

9085 

9090 

9096 

9101 

9106 

9112 

9117 

9122 

9128 

9133 

82 

9138 

9143 

9149 

9154 

9159 

9165 

9170 

9175 

9180 

9186 

83 

9191 

9196 

9201 

9206 

9212 

9217 

9222 

9227 

9232 

9238 

84 

9243 

9248 

9253 

9258 

9263 

9269 

9274 

9279 

9284 

9289 

85 

9294 

9299 

9304 

9309 

9315 

9320 

9325 

9330 

9335 

9340 

86 

9345 

9350 

9355 

9360 

9365 

9370 

9375 

9380 

9385 

9390 

87 

9395 

9400 

9405 

9410 

9415 

9420 

9425 

9430 

9436 

9440 

88 

9445 

9450 

9455 

9460 

9465 

9469 

9474 

9479 

9484 

9489 

89 

9494 

9499 

9504 

9509 

9513 

9518 

9523 

9528 

9533 

9538 

90 

9542 

9547 

9552 

9557 

9562 

9566 

9571 

9576 

9581 

9586 

91 

9590 

9595 

9600 

9605 

9609 

9614 

9619 

9624 

9628 

9633 

92 

9638 

9643 

9647 

9652 

9657 

9661 

mm 

9671 

9675 

9680 

93 

9685 

9689 

9694 

9699 

9703 

9708 

9713 

9717 

9722 

9727 

94 

9731 

9736 

9741 

9745 

9750 

9754 

9759 

9763 

9768 

9773 

95 

9777 

9782 

9786 

9791 

9795 

9800 

9805 

9809 

9814 

9818 

96 

9823 

9827 

9832 

9836 

9841 

9845 

9850 

9854 

9859 

9863 

97 

9868 

9872 

9877 

9881 

9886 

9890 

9894 

9899 

9903 

9908 

98 

9912 

9917 

9921 

9926 

9930 

9934 

9939 

9943 

9948 

9952 

99 

9956 

9961 

9965 

9969 

9974 

9978 

9983 

9987 

9991 

9996 

286 


ANALYTIC  GEOMETRY 


NATURAL   SINES,   COSINES,   AND   TANGENTS 


Deg. 

Eau. 

Sin. 

Cos. 

Tan. 

Deg. 

Rad. 

Sin. 

Cos. 

Tan. 

0 

0 

0 

1.0000 

0 

45 

.7864 

.7071 

.7071 

1.0000 

1 

.0175 

.0176 

.9998 

.0175 

46 

.8029 

.7193 

.6947 

1.0355 

2 

.0349 

.0349 

.9994 

.0349 

47 

.8203 

.7314 

.6820 

1.0724 

3 

.0524 

.0523 

.9986 

.0624 

48 

.8378 

.7431 

.6691 

1.1106 

4 

.0698 

.0698 

.9976 

.0699 

49 

.8552 

.7647 

.6561 

1.1504 

5 

.0873 

.0872 

.9962 

.0875 

50 

.8727 

.7660 

.6428 

1.1918 

6 

.1047 

.1045 

.9945 

.1051 

51 

.8901 

.7771 

.6293 

1.2349 

7 

.1222 

.1219 

.9925 

.1228 

52 

.9076 

.7880 

.6167 

1.2799 

8 

.1396 

.1392 

.9903 

.1406 

53 

.9250 

.7986 

.6018 

1.3270 

9 

.1571 

.1564 

.9877 

.1584 

54 

.9426 

.8090 

.6878 

1.3764 

lO 

.1745 

.1736 

.9848 

.1763 

55 

.9699 

.8192 

.6736 

1.4281 

11 

.1920 

.1908 

.9816 

.1944 

56 

.9774 

.8290 

.5592 

1.4826 

12 

.2094 

.2079 

.9781 

.2126 

57 

.9948 

.8387 

.5446 

1.5399 

13 

.2269 

.2260 

.9744 

.2309 

58 

1.0123 

.8480 

.5299 

1.6003 

14 

.2443 

.2419 

.9703 

.2493 

59 

1.0297 

.8672 

.5150 

1.6643 

15 

.2618 

.2588 

.9659 

.2679 

60 

1.0472 

.8660 

.5000 

1.7321 

16 

.2793 

.2756 

.9613 

.2867 

61 

1.0647 

.8746 

.4848 

1.8040 

17 

.2967 

.2924 

.9563 

.3057 

62 

1.0821 

.8829 

.4695 

1.8807 

18 

.3142 

.3090 

.9511 

.3249 

63 

1.0996 

.8910 

.4540 

1.9626 

19 

.3316 

.3256 

.9455 

.3443 

64 

1.1170 

.8988 

.4384 

2.0603 

20 

.3491 

.3420 

.9397 

.3640 

65 

1.1345 

.9063 

.4226 

2.1446 

21 

.3665 

.3584 

.9336 

.3839 

66 

1.1619 

.9135 

.4067 

2.2460 

22 

.3840 

.3746 

.9272 

.4040 

67 

1.1694 

.9205 

.3907 

2.3569 

23 

.4014 

.3907 

.9205 

.4246 

68 

1.1868 

.9272 

.3746 

2.4751 

24 

.4189 

.4067 

.9136 

.4452 

69 

1.2043 

.9336 

.3584 

2.6051 

25 

.4363 

.4226 

.9063 

.4663 

70 

1.2217 

.9397 

.3420 

2.7475 

26 

.4538 

.4384 

.8988 

.4877 

71 

1.2392 

.9456 

.3256 

2.9042 

27 

.4712 

.4540 

.8910 

.5096 

72 

1.2566 

.9511 

.3090 

3.0777 

28 

.4887 

.4696 

.8829 

.6317 

73 

1.2741 

.9563 

.2924 

3.2709 

29 

.6061 

.4848 

.8746 

.5643 

74 

1.2916 

.9613 

.2756 

3.4874 

30 

.5236 

.5000 

.8660 

.5774 

75 

1.3090 

.9659 

.2588 

3.7321 

31 

.5411 

.5150 

.8672 

.6009 

76 

1.3266 

.9703 

.2419 

4.0108 

32 

.5585 

.5299 

.8480 

.6249 

77 

1.3439 

.9744 

.2250 

4.3316 

33 

.6760 

.5446 

.8387 

.6494 

78 

1.3614 

.9781 

.2079 

4.7046 

34 

.6934 

.5692 

.8290 

.6745 

79 

1.3788 

.9816 

.1908 

5.1446 

35 

.6109 

.5736 

.8192 

.7002 

80 

1.3963 

.9848 

.1736 

5.6713 

36 

.6283 

.5878 

.8090 

.7266 

81 

1.4137 

.9877 

.1664 

6.3138 

37 

.6458 

.6018 

.7986 

.7536 

82 

1.4312 

.9903 

.1392 

7.1154 

38 

.6632 

.6157 

.7880 

.7813 

83 

1.4486 

.9926 

.1219 

8.1443 

39 

.6807 

.6293 

.7771 

.8098 

84 

1.4661 

.9945 

.1045 

9.5144 

40 

.6981 

.6428 

.7660 

.8391 

85 

1.4836 

.9962 

.0872 

11.4301 

41 

.7156 

.6561 

.7547 

.8693 

86 

1.5010 

.9976 

.0698 

14.3007 

42 

.7330 

.6691 

.7431 

.9004 

87 

1.6184 

.9986 

.0623 

19.0811 

43 

.7506 

.6820 

.7314 

.9326 

88 

1.5359 

.9994 

.0349 

28.6363 

44 

.7679 

.6947 

.7193 

.9667 

89 

1.5533 

.9998 

.0175 

57.2900 

1 

90 

1.6708 

1.0000 

0 

CO 

ANSWERS  TO   PROBLEMS 

Exercise  V 

2.  36°  53',  143°  T.  4.   27°  46',  152°  14',  etc. ;  9560  ;  .35i 

3.  24°  31',  204°  31'. 

Exercise  VI 

3.  (7.62,  -66°  44'),  (5,  36°  52').  8.    y  =  0;  x  =  yVS  ;  y  =  cx] 

4.  (1.73,  1),  (-  2.12,  -  2.12).  x^ -\- y^  =  25;  x^  -{-y^=  c\ 
6.   ^  =  90°;^  =  0;  rcos^  =  c;                9.    (4.21,  -  39°  19'). 

^  =  45°  ;  d  =  135°.  10.    (5.24,  3.57). 

Exercise  VII 

3.   7.62,  VxH^.  4.    11.40.  6.   x'^  +  y'^  =  25. 

Exercise  Vni 
1.  6.04.        2.    11.65.        3.    V(a-c)'^+(6-d)2.       4.  7.47.       6.    5. 

Exercise  IX 

1.  8.06.  2.   9.90.  4.    5.95. 

3.  AB  =  7.07,  BC  =  8.36,  6.    12.73,  14.87,  2.24. 

CA  =  7.70,  OA  =  3.91,  6.    5.99,  5.54,  6.40. 

OB  =4.27,  00  =  5.15. 

Exercise  X 

3.    -  1  :  5,  3  : 1,  -  7  :  3.  4.    r  =  2,  A:  =  -  V. 

Exercise  XI 

1-    (¥>-!)-  3    (a-nc     b-nd\ 

2.  (-29,  27.5),  (27,  -18).  '\l-n'    1-n) 
.      fxi  +  2x2     yi  +  2  y2\      f2xi+_X2     2  yi  +  ygN 

[       S        '  3       J'    V       3        '  3        j* 

287 


288  ANSWERS 

Exercise  XII 

2.  26°  34',  63°  26'.  .        ^q     _  6,  99°  25'.      12.       ^^    ■ 

3.  (-^,0).  ^-m^ 

6.  (12,  -  1),  or  (-  6,  -  19),  13.    87°  4'.  14.    -  .3332. 
or  (2,  9).                                          15.  (1,2,  -  4.56).    16.   (0,  -  am). 

7.  139°  24'.        8.  42°  60'.  ^^     m -\- n  ^    /Q^-a(m  +  w)\ 

'    1  —  ?w«      \  '      1  —  mil     ) 

Exercise  XIII 

1.    71.5.  2.   22.56.  4.   ^  nrg  sin  (^2  -  ^i)- 

3.   i(xiy2- XiVi).  5.   160. 

Exercise  XIV 

1.  185.        2.    1842  sq.  ft.  16.  114°  19'.  17.    -  4.871. 

3.  60305  sq.  ft.  18.  -  4.186. 

4.  (d)  41°  3',  38°  27',  100°  30'.  ^g      a  +  b  ^0.    3.154. 
12.  13.5.         IZ.   x  +  y  =  Q.                   '  1  -  ab 

14.   a;2  +  y2  _  4  a;  _  6  y  =  12. 

Exercise  XV 

1.   Sx-{-7y  =  Sl.  12.  x^-\-y-2±2rx±2ry-\-r^=0. 

4.    2x- 2/- 11  =  0.  14.  16x2  +  7  2/2  =  112. 

g     ^  .  y^l  15.  4x2 -5?/2  + 20  =  0. 

a      6       *  16.  x2  +  82/  +  16  =  0. 

6.  y  =  mx  +  b.  17.  63x2  +  143  y'^  -\Sxy  +  216  x 

7.  x2  +  i/2_4a;  +  8y  =5;  Inter-                      —  456 1/ —  1728  =  0. 
cepts,   X  =  5  or  -  1 ;  y  =  .58       18.  52  x2  -  80  y^  +  224  x?/  -  68  x 
or -8.58.                                                         +  496  !/- 1343  =  0. 

10.  (X  -  70^ -f  (2/ -  A;)2  =  r2.  19.    2/2  +  22  x  -  8  y  -  39  =0. 

11.  x^  +  y^  =  r^ 

Exercise  XVIII 

1.   x2  +  42,2=l8.              .  ^     x2=-^y,    e  =  45°,     new 

2.64x2-64^2  +  3  =  0.  2     origin  (1.77,  .93). 

3.  4x2  +  2/2:^=12.  g    Lines  x-2  2/=0,  and  x+2/=:0, 

6.   Lines  x  +  2  y  =  0  and  referred    to   If  axes    through 

2  X  —  3  2/  =  0,    referred  to                   ,q    _  -j^n 

II  axes  through  (1,  1).  ^    2  ^2  _^  ^./^  4^  ^  ^  450^ 

^  =  45°. 


ANSWERS 

Exercise  XIX 

1. 

2x  +  y  =  5. 

22.   3x  +  4y+75=0. 

3. 

7. 

8a;-3.y  =  24. 

3x  +  2y±5\/l3  =  0. 

^•^-^=i'-rj^-''> 

8. 

3x-y  =  n, 

27.    -  .5642  X  +  .8257  y  =  3, 

9. 

bx—  ay  =  0. 

.9780  X  +  .2088  y  =  S. 

10. 

y  -yi  =m(x-x{). 

31.   3x4-^+10=0. 

11. 

Bx  -Ay  +  Ab  =  0. 

33.   107  a;  +  134  y  -  187  =  0. 

20. 

y-2  =  6Mix-l). 

34.  _5x  +  5  2/=  12. 

21. 

Li,ix  +  Sy  +  18=0 

1;                 35.  'll8x  +  177y  =  486. 

Za,  11x-6y-hS9  = 

0.                36.    63  a;  +  147  y  =  536. 

289 


Exercise  XX 

1.   3.84.  2.  .383.  5.   1.06.  6.  3.13. 

w    mxi  —  y\  +  b  a  .        . 

7. ^'— —  •  8.  Xq  cos  a  +  2^0  sm  a  —p. 

±  Vw2  + 1 

Exercise  XXI 

1.   1.23  r  sin  0  -  .134  rcos0  =  1.         6.    (4.91,  102°  50'). 
4.    (8.94,  26°  34'). 

Exercise  XXII 

1.  (0,  0),r  =  5.  13.   x^  +  y^-h2x-6y  +  6  =  0. 

2.  (2,  -3),r  =  5.  16.   x^  +  y^ +  2x-6y  =  36. 

4.  (-.75,  1.75),  r  =  3.02.  16.  a;2  _|.  2/2-4x  -  lOy +20  =  0. 

6.  (1,  -2),r  =  0.  n.  x^  +  y^  +  6x-\-12y  =  85. 

7.  (-.5, -.5),r=.707.  18.  x^ -{- y^ -llx -17  y -h  S0=0. 

8.  No  locus.  19.  27 (x^  +  y^)- 66x  -\- 16y 

9.  (2,  -3),  r=5.10.  -250  =  0. 

12.    («,  ^V   r  =  iV^M::P.  20.   (x-H)2+(,  +  |)2=(H)«. 

V2'   2/'  2         ^  21.    (X -^«-)H  (!/  +  ¥)'=(¥)'• 

Exercise  XXIII 

1.  x2  =  ^y,  F(|,  V).  '  9-    19-44  ft,  17.78  ft.,  15ft., 

3.  x2  =  y,  F(-2,  0).  11.11  ft.,  6.11  ft. 

4.  x2  =  f  y,  F(l,  i).  10.    5  y  =  x2  +  X  -  2. 
6.  i/2  =  -tx,  F(f,^). 

12.   2  A2y  =  (yj  ^.  yg  _  2  y2)x«  +  /i(y3  -  yi)x  +  2  ^2^2- 
u 


0 

ANSWERS 

Exercise  XXIV 

1. 

x2=4y,  F(3,  -2).                      12.  (V2-l)x2-(V2  +  l)y2=10, 

2. 

??  +  l^'  =  1,0(2,3).                                 ^=67°  30'. 

4       9'^^                       13.   8  x-2 +  28^2  =  13, 

3. 

The  lines  Sx-y  +  'J  =  0,                       c^-  1,  1),  ^  =  tan-i ^. 

3a;  +  «/  +  5  =  0.             14.   The  lines  2  x  +  7  ?/ =  0, 

4. 

^^_l!-l    of-    -3V                            7x-2y=0,    referred    to 
1      16       '      V2'        /                             II  axes  through  (-3^,  it). 

8. 

^^y!^l   .^tan-12.                 15-  x^  -  t,2  =  2,  0(- 1,  -  2), 

If                                                          ^  =  45°. 

9. 

9%-i>'''  =  ''°'"-          "■.-*  =  4(-^)- 

10.  i/2=4x,  0  =  tan-if  ;  19.   x  =  6  and  y  =  a. 

F(-3,-3).  23.  xy-4x-2y  +  12=0. 

11.  x2-y2  =  i6,  ^  =  45°. 


Exercise  XXVIII 
8.    ±3V5.  9.    ±  r  Vl  +  m2.  10.  6. 

Exercise  XXX 

1.  y  =  VSx±S.  16.  y  =  :^a;  +  3V2, 

2.  y  =  x±10.  ^ 

3.  x  +  2y  +  6  =  0.  y  =  _^2^_3V2. 
6.  x-2«/+6=0,3x-2y  +  2  =  0.  4 

12.  2x  +  2y+P  =  0.  17.   y  =  2x-7±2VlO. 

Exercise  XXXI 

1.  .007651,. 030301, 3.003001.  3.  f ,  3f ,  0,  f . 

2.  -1,  -hh  4-  4x-4y  =  5. 

Exercise  XXXII  ' 

1.   x  +  y  +  1  =  0,  x-y  =  3.  2.  3x-4y4-26=0,4x+3y=0. 

3.  Tangents,  2?/oy- 3xo2x  +  Xo^  =  0,  y  =  0,  Sx-2y=l,  Sx-y=A. 
6.   Tangents,  6x-i/  =  6,  6x  +  y4-30  =  0. 

8.    12°  6',  36°  52'.  9.   73°  41'.  10.  (-2,  -  9). 


ANSWERS  291 


Exercise  XXXIII 


2.  2aa;-^.  6.  '        4-1. 

4.      -^«  8       ^^  -^ 


9.    (x  -f  ay-^(x  +  6)«-i[rc(n  +  m)  +  ma  +  w6]. 

10     ~  ^^ .  14.   8  a;  —  y  =  4. 

s"+i  *  15.  x-y+1  =0. 

11.  2  anx(ax^  +  b)^-\  16.  4a;  -  Sj^  +  25  =  0. 

12.  -2  an-  ^^-^^^^^-  21.  l(y+yo)  =  axoa;+^(x+a;o)  +  c. 

(x  —  a)"+i  2  2 

13.  y  =  mz+b.  ^    i(=c+^„)  =  «y„,+| (,+,„)+<. 


Exercise  XXXIV 


1. 

(6  -  a)  sin  2  x.                                5. 

—  a  sin  2  (ax  +  6) . 

2. 

24tan2  2x(l+tan2  2a-0.                 g 

2sinx           -     12  sin  3  a; 

3. 
4. 

1  cos «  Vsin  t. 
-  2  sin  2  X. 

cos^  a;           '      cos^  3  x 

8 

2(1  +  sin  t)  sec2  2  «  -  cos  t  tan  2  t 

(1  -f  sin  0^ 

9.   .-^l5^(l  +  3cos2x).  1*-   a^cosx  +  sinx. 

2  cost  X  15.x  sec^  x  -f  tan  x. 

10.   wm(tan«-imx  +  tan«+iwx).  ^6.  siux  +  xcosx. 

2(sin4x  +  cos4x)  1^.   4csc4x  (1  -  2  csc24x). 


11. 
12. 

13.   tan4^. 


sin^xcosSa;  18.  —  mng  ^*^^"~  ^^ » 

cosx  sin»»+igx 

3    ■  19.  (x  + 1)  sin x+ (x— 1)  cosx 

20.  abn  sin  «-i  6«  cos  bt. 


Exercise  XXXVII 

1.   y^-6y  +  Sx  =  23.  2     ?!+(y±I)!^l 

3.   3  x2  -  2/2  «  16  X  -f  8  2/  =  0.  '6  15 


292  ANSWERS 


Exercise  XLI 


1.   No  locus.  2.   Hyperbola.         3.   Two  intersecting  lines. 

4.  Two  parallel  lines.        5.   One  line.  6.   Ellipse.       7.   A  point. 

8.  No  locus.  9.   Parabola.  10.    Hyperbola. 


Exercise  LIII 

2.  x=~(h-  be)  cos  e,        y  =^{h-  bd)  sin  0,  z  =  bd. 

h h 

3.  a;  =  y/a^  -  b'^ff^  •  cos  6,     y  =  Va^  —  6^^  •  sin  0,  z  =  be. 

A.  r  =  ^{h-b0).  6.   r  =  VcF^^W\ 

h 


Exercise  LV 

1.  x^-^-y^y  +  z^z^r"^.  8.  p^p+v^p  =  B{t  +  Q. 

•  a2  5-2  "^  c-2  ■  14  2a;-V6^2y-V3^2g-l 
4.  a;oic  +  2/oy  =P(^+^o).  *  2V3  V6  --3V2 
g    ^  ,  |/oy _g  +  gp  three  other  answers. 

a2        ft2         2c    '  ^g     x-\^y-\  ^z-2 

a    ^(y^_M  =  l  3           -1           4 

*  a2        52         •   . 


INDEX 


The  numbers  refer  to  the  pages. 


Abscissa,  7. 

Addition  of  segments,  3. 

Angle,  between  two  lines,  21,  27,  241. 

between  two  planes,  252. 
Area,  of  a  triangle,  30,  31,  33. 

of  a  polygon,  35. 
Asymptote,  59. 

of  the  hyperbola,  108. 
Axes,  of  the  ellipse,  103,  198. 

of  the  hyperbola,  109,  200. 
.Axis  of  the  parabola,  95. 

Cardioid,  138,  140. 
Center,  of  ellipse,  103. 

of  hyperbola,  109, 
Change  of  sign  oiAx  +  By  +  C,  81. 
Circle,  equation  of,  88. 

through  three  points,  89. 
Circular  measure  of  an  angle,  172. 
Concavity,  183. 
Cone,  262. 
Conic  Sections,  192,  264. 

classification  of,  196. 

polar  equation  of,  202. 

rectangular  equation  of,  196. 
Conjugate  diameters,  213. 
Conjugate  hyperbola,  109. 
Continuity  of  functions,  161. 
Coordinate  planes,  233. 
Coordinates,  Cartesian,  6. 

rectangular,  9. 

polar,  9. 

rectangular,  in  space,  233. 

polar,  in  space,  236. 

spherical,  238. 
Cycloid,  132,  140. 

construction  of,  133. 
Cylinders,  equations  of,  243. 

Derivative  curves,  182,  185. 


Derivatives,  159. 

partial,  271. 
Diameter,  of  parabola,  208. 

of  ellipse,  213. 

conjugate,  213. 
Differentiation,  161. 

formulas  of,  162,  169,  175. 
Direction  cosines  of  a  line,  237. 
Directrix,  of  parabola,  92. 

of  conic,  192. 
Discontinuity,  161. 
Distance,   between    two    points,    18, 
20,  235,  236. 

from  a  point  to  a  line,  83. 

to  a  plane,  251. 

Eccentric  angle  of  ellipse,  130. 
Eccentricity  of  a  conic,  192. 
Ellipse,  definition  of,  45,  100. 

construction  of,  131. 
Ellipsoid,  247,  257. 
Elliptic  paraboloid,  248,  260. 
Empirical  equations,  223. 
Epicycloid,  138. 
Equation  of  a  locus,  41. 
Exponential  function,  123. 

Focal  radii,  of  ellipse,  208. 

of  hyperbola,  209. 
Foci  of  ellipse,  100. 
Focus  of  a  parabola,  92. 
Function  and  variable,  38. 

General  equation  of  second  degree,214. 

Graph  of  a  function,  39. 

Graphical  solution  of  equations,  143. 

Helix,  268. 

Hyperbola,  definition  of,  46,  105. 
equilateral,  92. 


293 


294 


INDEX 


Hyperbolic  paraboloid,  261. 
Hyperboloid,  of  one  sheet,  258. 

of  two  sheets,  259. 
Hypocycloid,  134. 

construction  of,  135. 

of  four  cusps,  137. 

Inclination  of  a  line,  22. 
Increments,  153. 
Intercepts,  48. 
Intersection,  of  lines,  79. 

of  curves,  142. 
Involute  of  circle,  139. 

Latus  rectum  of  a  conic,  206. 

Limit  of  ^— :,  146. 
sin  0 

Locus  of  an  equation,  51. 

Logarithmic  curve,  122. 

Maxima  and  minima,  178. 

Normal,  to  a  curve,  158. 
to  a  surface,  276. 

Ordinate,  7. 

Parabola,  47,  92,  97. 

parameter  of,  95. 
Parabolic  arch,  99. 
Parallel  lines,  condition  for,  28. 
Parametric  equations  of  loci,  129. 
Periodic  functions,  118. 
Perpendicular  lines,  condition  for,  28. 
Plane,  equations  of,  249,  250. 
Plotting  in  polar  coordinates,  125. 
Projections,  15,  239. 
Property  of  reflection,  of  parabola,  206. 


Property  of  ellipse,  209. 
of  hyperbola,  210. 

Quadric  surfaces,  257. 

Radical  axis  of  circles,  91. 

Ratio  into  which  a  point  divides  a  line, 

23,  26,  235. 
Rotation  of  axes,  65. 

Sine  curve,  1 17. " 
Slope,  of  a  line,  22. 

of  a  curve,  155. 
Space  curves,  268. 
Standard  equations  of  second  degree, 

88. 
Straight    line,    equations    of,    70-74, 

86,  253. 
Subnormal  of  parabola,  205. 
Subtangent  of  parabola,  205. 
Subtraction  of  segments,  4. 
Surfaces  of  revolution,  244. 
Symmetry,  55. 

Tangent  plane  to  a  surface,  273. 
Tangents,  slope  equations  of,  148, 149. 

contact  equations  of,   156,   159. 

to  space  curves,  277. 
Transformation  of  coordinates,  64. 
Translation  of  axes,  64. 
Trigonometric  functions,  11. 

Variable,  dependent  and  independent, 

38. 
Vertex  of  a  parabola,  95. 
Vertices,  of  ellipse,  103. 

of  hyperbola,  109. 

of  conies,  193. 


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